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Shannon entropies of discrete probability distributions

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Numbers
distribution
shape
unit 
$H(X)$
Bernoulli$(p)$ on $\{0,1\}$
1/2
nats:
0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
comment: In bits this entropy is exactly $1$.
Bernoulli$(p)$ on $\{0,1\}$
1/3
nats:
0.6365141682948128184504238226170746592638238015825792823209076606418985452388124898030402034827862971
Bernoulli$(p)$ on $\{0,1\}$
1/3
bits:
0.9182958340544895147870722772811498420931477410258143937890859878744315611276918958556138082514215754
Bernoulli$(p)$ on $\{0,1\}$
1/4
nats:
0.5623351446188083502880303152244588576653823503534484194403392687586665240254327060565148378825383085
Bernoulli$(p)$ on $\{0,1\}$
1/4
bits:
0.8112781244591328639096957920391376184301391942306392046581855090941763291542310781082896438114338184
Bernoulli$(p)$ on $\{0,1\}$
1/5
nats:
0.5004024235381878795331879388931051306048011392921093153195598762847491925561462196607525548989097107
Bernoulli$(p)$ on $\{0,1\}$
1/5
bits:
0.7219280948873623478703194294893901758648313930245806120547563958159347766086252158501397433593701551
Bernoulli$(p)$ on $\{0,1\}$
1/10
nats:
0.3250829733914482395065500282238179392356184845478239629108781004200828818838563398634888464248805385
Bernoulli$(p)$ on $\{0,1\}$
1/10
bits:
0.4689955935892812212535893303833204600971654591781147032344016176419579665787798033100348885068113193
Bernoulli$(p)$ on $\{0,1\}$
2/5
nats:
0.6730116670092564359967193424894015895069069658307659492235592874943249629886144982636190944063728825
Bernoulli$(p)$ on $\{0,1\}$
2/5
bits:
0.9709505944546686389980760631207002706089427484090919757813048030912758399320100783367714584085172098
Bernoulli$(p)$ on $\{0,1\}$
1/6
nats:
0.4505612088663046886451779140255459064516562302925732709281677669024054254319222119543675167325011007
Bernoulli$(p)$ on $\{0,1\}$
1/6
bits:
0.6500224216483542248951394193733246955391215801719972170767889913611525806205042159804973554519464462
Bernoulli$(p)$ on $\{0,1\}$
1/7
nats:
0.4101163182884090186946578648311492101602355648535714405833549986825330783049518256623907141791708616
Bernoulli$(p)$ on $\{0,1\}$
1/7
bits:
0.5916727785823273804816216509908452297040428479440141604295037344636652250384570757644777349547065776
Bernoulli$(p)$ on $\{0,1\}$
2/7
nats:
0.5982695885852572348427440421507095390972265810014184573444473676007681223981152314501733493794807514
Bernoulli$(p)$ on $\{0,1\}$
2/7
bits:
0.8631205685666310018203125818822663973090042023771546322096085842017960084274606894620468967706606743
Bernoulli$(p)$ on $\{0,1\}$
3/7
nats:
0.6829081047004716551734109316670383498447314798175339901494582818147753119787286070065508013860792746
Bernoulli$(p)$ on $\{0,1\}$
3/7
bits:
0.9852281360342514582475097698256237334582490226693631863391120149812787512360393168454550813481729671
Bernoulli$(p)$ on $\{0,1\}$
1/8
nats:
0.3767701612564367862845127138617474407940512646966372224600736472847984860010235375033887947248684765
Bernoulli$(p)$ on $\{0,1\}$
1/8
bits:
0.5435644431995964059882768474221480424391017022796268142823697414384692572455811368143716257260655630
Bernoulli$(p)$ on $\{0,1\}$
3/8
nats:
0.6615632381579820598530048172622152902801905974794111417661247211947586386348196813461503991258692726
Bernoulli$(p)$ on $\{0,1\}$
3/8
bits:
0.9544340029249649645358982525886999492995499764749567197948700071621289291967247791478074823061105623
Bernoulli$(p)$ on $\{0,1\}$
1/9
nats:
0.3488320958430318910112048166232472277603140906848182258142419752926055945180320254649293043036810108
Bernoulli$(p)$ on $\{0,1\}$
1/9
bits:
0.5032583347756456962408112212289663508529621487182954542448386424155297889220504583778942831695098175
Bernoulli$(p)$ on $\{0,1\}$
2/9
nats:
0.5297061990576545211713867575096501600049151850017723670852674374361167771923430108888608269488043927
Bernoulli$(p)$ on $\{0,1\}$
2/9
bits:
0.7642045065086202793415017522708757219099414396335192891625006348052802398070114688795579504815680958
Bernoulli$(p)$ on $\{0,1\}$
4/9
nats:
0.6869615765973233440858622918676791046025357993983177209662020524618725959599016506363897311344366715
Bernoulli$(p)$ on $\{0,1\}$
4/9
bits:
0.9910760598382221696461893159570829198169447081490840031033073114066771352505920051278166479698597313
Bernoulli$(p)$ on $\{0,1\}$
3/10
nats:
0.6108643020548934630256709631973806854608950105746453085913464959200184177853213027225515296339686028
Bernoulli$(p)$ on $\{0,1\}$
3/10
bits:
0.8812908992306926182248192242427636571881684325405377453439263926043807140667825565449529014647961329
Bernoulli$(p)$ on $\{0,1\}$
1/11
nats:
0.3046360973492381040455877100702527474570691370021690151882696173418712514792799128605117942848181566
Bernoulli$(p)$ on $\{0,1\}$
1/11
bits:
0.4394969869215133035899817471899837079170211683866947876266594677118444591667227019366369871420185296
Bernoulli$(p)$ on $\{0,1\}$
2/11
nats:
0.4741393130578373564302273500086005889302677348892352989034897069067841219246593644444355952171788848
Bernoulli$(p)$ on $\{0,1\}$
2/11
bits:
0.6840384356390416865477902293566386716417170403667990632942973019318062469656134322184868857845742744
Bernoulli$(p)$ on $\{0,1\}$
3/11
nats:
0.5859526183035508622256429756229643227531182268361104066639855974256709141924639789694576977885216307
Bernoulli$(p)$ on $\{0,1\}$
3/11
bits:
0.8453509366224364794392524256491157290414639599474550548248691035787583757761023811125057152183309682
Bernoulli$(p)$ on $\{0,1\}$
4/11
nats:
0.6554817739013927612369139256226137859977437464869562341111885159023728952748692040422805165402302288
Bernoulli$(p)$ on $\{0,1\}$
4/11
bits:
0.9456603046006400969181098448509915350225782182487693907908887304088537158986228158472161179967663528
Bernoulli$(p)$ on $\{0,1\}$
5/11
nats:
0.6890092384766585551638835037455700421884385880791792802462508440254624939434084477998234460638345449
Bernoulli$(p)$ on $\{0,1\}$
5/11
bits:
0.9940302114769564536471784275318066012593184882928781237665927451512157788282658568563657023500477407
Bernoulli$(p)$ on $\{0,1\}$
1/12
nats:
0.2868359830561606448395945333708436492952595437672942160257006192545890485708647444332089264708209051
Bernoulli$(p)$ on $\{0,1\}$
1/12
bits:
0.4138168503036336966043228177825062966151855091508603284190178126252951597176750725385801172384294608
Bernoulli$(p)$ on $\{0,1\}$
5/12
nats:
0.6791932659915257261679373239861124820411908366752918825778068103797853723411005899466525020914862353
Bernoulli$(p)$ on $\{0,1\}$
5/12
bits:
0.9798687566511528071666237466086692971088691287853236816211839839101049090378188137943033323357277195
Bernoulli$(p)$ on $\{0,1\}$
1/20
nats:
0.1985152433458725564266475158490799153010913247548558525429456858721381646860597715634100674610640849
Bernoulli$(p)$ on $\{0,1\}$
1/20
bits:
0.2863969571159561287664759777278974743059992018461232982023079014945978288134335733875670918958024613
Bernoulli$(p)$ on $\{0,1\}$
1/13
nats:
0.2711893730418441419952940755601996748374302587202748456719184159333586330247193503977201322344253683
Bernoulli$(p)$ on $\{0,1\}$
1/13
bits:
0.3912435636292556852087459367664025716963000254595163906562303062664568225151558578191176303567657498
Bernoulli$(p)$ on $\{0,1\}$
1/14
nats:
0.2573186405438316453300608120192747361076932009504955487623850423863277222062402453796310666752642607
Bernoulli$(p)$ on $\{0,1\}$
1/14
bits:
0.3712323266408756727877861382791856988431928243030346548201470215019553184172409220752970781948526671
Bernoulli$(p)$ on $\{0,1\}$
1/15
nats:
0.2449300267946353591082586962407807996413460390786251605726100763094313618526203470031292623615741660
Bernoulli$(p)$ on $\{0,1\}$
1/15
bits:
0.3533593350214213623782203440208312632263542831486636077450367745580668787982703243077499523852616644
Bernoulli$(p)$ on $\{0,1\}$
1/16
nats:
0.2337916587064593007976742013182875121397268698554580411883367019313807870104038016386880271354450318
Bernoulli$(p)$ on $\{0,1\}$
1/16
bits:
0.3372900666170138787586952749026187331643945618277546820213977652902815583722027077758560453648827526
Bernoulli$(p)$ on $\{0,1\}$
1/17
nats:
0.2237180760658337389140725135599912204804378008765485364899718197895165872091394103959035193451031827
Bernoulli$(p)$ on $\{0,1\}$
1/17
bits:
0.3227569588973982317834777755162867069524437529330559822442561232102259208120879506502843283142725085
Bernoulli$(p)$ on $\{0,1\}$
1/18
nats:
0.2145591551764050608609399006452751382038450714525507473648997021017725534838108840472274612551300814
Bernoulli$(p)$ on $\{0,1\}$
1/18
bits:
0.3095434291503251440008599887969177942867652709481870265697078593836497525995229494304035286504746706
Bernoulli$(p)$ on $\{0,1\}$
1/19
nats:
0.2061920506332318042332902363374270323600812349778880376631583321024042940997537261691211701720613920
Bernoulli$(p)$ on $\{0,1\}$
1/19
bits:
0.2974722489192895710390666870057084060959644814020962158232565433130106713319433999183871543273777453
binomial $\operatorname{Binom}(n,p)$
2, 1/2
nats:
1.039720770839917964125848182187264852113250201540382881181020014240090432954542073408794990494628031
comment: In bits this entropy is exactly $3/2$.
binomial $\operatorname{Binom}(n,p)$
2, 1/3
nats:
0.9649629230074277216043000356971819549385364323383784516992908726200665918244273282256967060782753997
binomial $\operatorname{Binom}(n,p)$
2, 1/3
bits:
1.392147223664534585129700110117855239741851037607184343133727531304418677810939347266783172058398706
binomial $\operatorname{Binom}(n,p)$
2, 1/4
nats:
0.8647400965276372095445985849021015023024521503218011185854235339573104398122298937608309281414196092
binomial $\operatorname{Binom}(n,p)$
2, 1/4
bits:
1.247556248918265727819391584078275236860278388461278409316371018188352658308462156216579287622867637
binomial $\operatorname{Binom}(n,p)$
2, 1/10
nats:
0.5253994542821063233179982745851640962176469449108019800800337991313549118131676309179222939904057133
binomial $\operatorname{Binom}(n,p)$
2, 1/10
bits:
0.7579911871785624425071786607666409201943309183562294064688032352839159331575596066200697770136226387
binomial $\operatorname{Binom}(n,p)$
3, 1/2
nats:
1.255482325178753659705262436682635425740882484713703673561019278252060145995127421662378164878956996
binomial $\operatorname{Binom}(n,p)$
3, 1/2
bits:
1.811278124459132863909695792039137618430139194230639204658185509094176329154231078108289643811433818
binomial $\operatorname{Binom}(n,p)$
3, 1/3
nats:
1.177134312439031994421107976569540174693144366199238212472926759500699440237364824826710107239204165
binomial $\operatorname{Binom}(n,p)$
3, 1/3
bits:
1.698245835016031090058724202544905187106233617949122474396756193929229198186836645885321108142205898
binomial $\operatorname{Binom}(n,p)$
3, 1/4
nats:
1.069036021480613349454265499904455864131933612285048691720252243604909032140830574303135651564890626
binomial $\operatorname{Binom}(n,p)$
3, 1/4
bits:
1.542292966721748239661359220146766069113021978364897017468195659103161234328366542906086164292876819
binomial $\operatorname{Binom}(n,p)$
3, 1/10
nats:
0.6786236022339551018429338707023718774520330030313295367642668311781251864825445985345902854749339516
binomial $\operatorname{Binom}(n,p)$
3, 1/10
bits:
0.9790469055731314947682584762840509229263464874573742233801516361997773782318625980490889372925501326
binomial $\operatorname{Binom}(n,p)$
4, 1/2
nats:
1.407531740719315302947017354981766351955378893512138997666274649806097897713470265887785325310449522
binomial $\operatorname{Binom}(n,p)$
4, 1/2
bits:
2.030639062229566431954847896019568809215069597115319602329092754547088164577115539054144821905716909
binomial $\operatorname{Binom}(n,p)$
4, 1/3
nats:
1.330575170632721091420485089014089007531939806809174743231983914239041097966424457632203232163773415
binomial $\operatorname{Binom}(n,p)$
4, 1/3
bits:
1.919614200201813017729897323263517933678324966762275532552170325460857633806266280946717808338598427
binomial $\operatorname{Binom}(n,p)$
4, 1/4
nats:
1.221565833659991821839584204247365512438242176318701821881851540630437905895109220322547363651682035
binomial $\operatorname{Binom}(n,p)$
4, 1/4
bits:
1.762346970340662573613385109667557928904033425299924094942856710809442409191551803151115037567701035
binomial $\operatorname{Binom}(n,p)$
4, 1/10
nats:
0.8040182879587177743067798687690621806963613112249071209060917309992707804463660403917509940648632332
binomial $\operatorname{Binom}(n,p)$
4, 1/10
bits:
1.159953196822076694595705608857417958062934856498604233399456891557134492444313387101556722946226189
binomial $\operatorname{Binom}(n,p)$
5, 1/2
nats:
1.523670872042791627512178656479971573275061818199381372484867643276554295141485226434901544896975873
binomial $\operatorname{Binom}(n,p)$
5, 1/2
bits:
2.198192411043097798871575534853696710126720569039455676198665878922561146929413860140493990600590480
binomial $\operatorname{Binom}(n,p)$
5, 1/3
nats:
1.449403667489066084860237160194506660642426413181952575817963577494328849424632422492436763754927532
binomial $\operatorname{Binom}(n,p)$
5, 1/3
bits:
2.091047483332751717751134718945782391817118982762150452354899720765794418445820403852022349465059595
binomial $\operatorname{Binom}(n,p)$
5, 1/4
nats:
1.342052132566523583882733760066268972567977079209492375936745016871964149110500293925528073561723941
binomial $\operatorname{Binom}(n,p)$
5, 1/4
bits:
1.936171956268181281131634082264316669128656433497753760202343572095462577651304151905599586420148857
binomial $\operatorname{Binom}(n,p)$
5, 1/10
nats:
0.9102051201901215240709433923248535557782431572829811318473861097727732305734505727373294795585357283
binomial $\operatorname{Binom}(n,p)$
5, 1/10
bits:
1.313148413090031224815050845540697023469178840447007755535585344123164541872666990659542217628394518
binomial $\operatorname{Binom}(n,p)$
6, 1/2
nats:
1.617363315627128389966918714250924359837365210400365332805878227583171449896471574077313577437113687
binomial $\operatorname{Binom}(n,p)$
6, 1/2
bits:
2.333362034770989421647296763745659341231972269151355700908133886226205243784463743461831763835496657
binomial $\operatorname{Binom}(n,p)$
6, 1/3
nats:
1.545763808367196802452635494910837817042923694939978260573813180304124171241075868843431470972016842
binomial $\operatorname{Binom}(n,p)$
6, 1/3
bits:
2.230065780716992787662033985614445004118874234538538576705863965704691556000274400096243012328794548
binomial $\operatorname{Binom}(n,p)$
6, 1/4
nats:
1.440856527920919460432077525675939770006411968478258662503947525147001278635554779155403036938158925
binomial $\operatorname{Binom}(n,p)$
6, 1/4
bits:
2.078716567464000746382205519487883111425618649525581118175810664471000279725884770494286511045427382
binomial $\operatorname{Binom}(n,p)$
6, 1/10
nats:
1.002110637862436226051932947291347991489534602628447445848698273558356523695895103326601722383690184
binomial $\operatorname{Binom}(n,p)$
6, 1/10
bits:
1.445740047666212632827261301962397665583707336495933700840054299282351197378580422461225025509430387
binomial $\operatorname{Binom}(n,p)$
7, 1/2
nats:
1.695881445361670996358688139657080817963974579938456878433621993734205727526313416475057320297031337
binomial $\operatorname{Binom}(n,p)$
7, 1/2
bits:
2.446639751158890300705722424365029003381095644480167295888177215385140727448412960670499625959670500
binomial $\operatorname{Binom}(n,p)$
7, 1/3
nats:
1.626535701413680573567114364616051680458047647758885429942065524978065585811395039799714365243110877
binomial $\operatorname{Binom}(n,p)$
7, 1/3
bits:
2.346594990258368670950901543501773251061677141641386111512897412398032459891020490562429095001725405
binomial $\operatorname{Binom}(n,p)$
7, 1/4
nats:
1.524134585535896514630342112619284833959840272525586170043598391326785046542518668438653140103310558
binomial $\operatorname{Binom}(n,p)$
7, 1/4
bits:
2.198861408199993518043579140327136547480577967197673953445820564457696215241968592589016810428495319
binomial $\operatorname{Binom}(n,p)$
7, 1/10
nats:
1.082910859190992303564912194062541418212784813567036515428464073105962690306456366723661768263342398
binomial $\operatorname{Binom}(n,p)$
7, 1/10
bits:
1.562310126279651136284761267845191382640285092960968752337467915359471802172427997368855193733764657
binomial $\operatorname{Binom}(n,p)$
8, 1/2
nats:
1.763503329079247483278167569280108451044816166728980793826670179612784419080308747149677173743189448
binomial $\operatorname{Binom}(n,p)$
8, 1/2
bits:
2.544197507453808039366753681385027168878505724641944729066295398297453920344509000406247266259116884
binomial $\operatorname{Binom}(n,p)$
8, 1/3
nats:
1.695948272579713645438916984778606143121827492750268957935254309381349332340629337402011935267559686
binomial $\operatorname{Binom}(n,p)$
8, 1/3
bits:
2.446736162454956835993684719249583682080500428545900096086412823157010803575376940556707934050622086
binomial $\operatorname{Binom}(n,p)$
8, 1/4
nats:
1.595837294114369676694653248364852863263195544007878491596002442978044237500231091987077258496754252
binomial $\operatorname{Binom}(n,p)$
8, 1/4
bits:
2.302306550284463283861732180402866013381074617211053393782376774567872445793293006028539101717521166
binomial $\operatorname{Binom}(n,p)$
8, 1/10
nats:
1.154807284671601414243057134152881600841410779339380788319558607305643928509168452586671337438990939
binomial $\operatorname{Binom}(n,p)$
8, 1/10
bits:
1.666034742778168807811173194784590666177836565898479697157622116220670346281198911848700228831307049
binomial $\operatorname{Binom}(n,p)$
9, 1/2
nats:
1.822926834573997996266005095054228498151173787287080337775102112428642575344556221564822388534986430
binomial $\operatorname{Binom}(n,p)$
9, 1/2
bits:
2.629927504143322672413647083045100380678529183086380873725070730901268658978682520316241046324805250
binomial $\operatorname{Binom}(n,p)$
9, 1/3
nats:
1.756760907085107480677741667209285009718550286131301967604111963714379057904661180339680765544085236
binomial $\operatorname{Binom}(n,p)$
9, 1/3
bits:
2.534470248679281582119762935747840009911144538871508667020145815129613514165255459507535007280231807
binomial $\operatorname{Binom}(n,p)$
9, 1/4
nats:
1.658638696541375776155122971383506143823243886117256952747286024261726676008628495525784385418476966
binomial $\operatorname{Binom}(n,p)$
9, 1/4
bits:
2.392909822126777094291591711593222580537134956839141917193345974765601930175699187383030443367880770
binomial $\operatorname{Binom}(n,p)$
9, 1/10
nats:
1.219402874428558693918021801052092418844003646225019721269900663341728088948770709943679396920616172
binomial $\operatorname{Binom}(n,p)$
9, 1/10
bits:
1.759226479783828996260906124226044676266477424798807067541961685798568494318811989675353300808256882
binomial $\operatorname{Binom}(n,p)$
10, 1/2
nats:
1.875953605246800506692960297874478945330495958490238702822337421692753792599613674647690156903157003
binomial $\operatorname{Binom}(n,p)$
10, 1/2
bits:
2.706428963227331175844751481882337766996210323030402125814615204046131438887057606287973484258781968
binomial $\operatorname{Binom}(n,p)$
10, 1/3
nats:
1.810859198431020054279542127351776025378825695322652407590746420261968170471985282958591497206964644
binomial $\operatorname{Binom}(n,p)$
10, 1/3
bits:
2.612517585324595977735927761873749294940512965245128714879507784562360572609354003175446971683354461
binomial $\operatorname{Binom}(n,p)$
10, 1/4
nats:
1.714421065073478753632386779898649184044511456784000365870797830659595092711955503592799509206191304
binomial $\operatorname{Binom}(n,p)$
10, 1/4
bits:
2.473386768577082625068177223842009060145196262291783597879890570978942868770985238050624138376266954
binomial $\operatorname{Binom}(n,p)$
10, 1/10
nats:
1.277907356882020645932289901991356735574852853234219885785862127161218366575042463805581896636647146
binomial $\operatorname{Binom}(n,p)$
10, 1/10
bits:
1.843630606489213929328267185673058019830700398368619021117294450784011288278862618143965530651660796
binomial $\operatorname{Binom}(n,p)$
11, 1/2
nats:
1.923844833782542271472213089034851963857391544620839267063348853614451966369653662877243806019740297
binomial $\operatorname{Binom}(n,p)$
11, 1/2
bits:
2.775521401137925832314412034239436498091445501665986825685374835748906653045061574574979829949804086
binomial $\operatorname{Binom}(n,p)$
11, 1/3
nats:
1.859580604820305004750806980237817064301792910855838438628683118273423130092748622415911280676739561
binomial $\operatorname{Binom}(n,p)$
11, 1/3
bits:
2.682807716707553232362653582176159181465032329096729664617625434077764300886698081489518821176246733
binomial $\operatorname{Binom}(n,p)$
11, 1/4
nats:
1.764550932701843141139175916303753608398776163522874606429765081601829937030137982792276079136346058
binomial $\operatorname{Binom}(n,p)$
11, 1/4
bits:
2.545708880004944108174272669629846281496292419783576004978783940433562725591435995945144592067025673
binomial $\operatorname{Binom}(n,p)$
11, 1/10
nats:
1.331259000346267769842474684232134774601793098000503901799583421610676655525643797237299959021451493
binomial $\operatorname{Binom}(n,p)$
11, 1/10
bits:
1.920600757938359331089985155711253438116021159164284871301768515933466162195532703091429391981824952
binomial $\operatorname{Binom}(n,p)$
12, 1/2
nats:
1.967518195598394159277750416068395856704945619749199080800889913206479323649741324501279707718827392
binomial $\operatorname{Binom}(n,p)$
12, 1/2
bits:
2.838528743648604764756779742096431109094410660387833873506433244715359233431948193089327201629708826
binomial $\operatorname{Binom}(n,p)$
12, 1/3
nats:
1.903903781863056105409291068057162679085102764541849776768075627261607167059004648547929421338954977
binomial $\operatorname{Binom}(n,p)$
12, 1/3
bits:
2.746752544423573795726130518044522052930891027372741496709309054115823672556400004595765997712479090
binomial $\operatorname{Binom}(n,p)$
12, 1/4
nats:
1.810045979409239726339332418954955382546802233293567463171829125890985927248335542434358522003305778
binomial $\operatorname{Binom}(n,p)$
12, 1/4
bits:
2.611344358274716924694482053881504731778124439454574502591614414498090075074718813518813822298630057
binomial $\operatorname{Binom}(n,p)$
12, 1/10
nats:
1.380201408880747723023243158553807751568015980108189871924766439908903032129074241657233853598504047
binomial $\operatorname{Binom}(n,p)$
12, 1/10
bits:
1.991209728020215238778524813935511816710832896597269062425522309603763101462632575164785252877037141
binomial $\operatorname{Binom}(n,p)$
13, 1/2
nats:
2.007662963297866686739242088094012158482642197888944467558604312551021421341168047688690578401498435
binomial $\operatorname{Binom}(n,p)$
13, 1/2
bits:
2.896445400926273220223533311152290911410544464605119678836341712436857857689912763969120828908633311
binomial $\operatorname{Binom}(n,p)$
13, 1/3
nats:
1.944564116523508037302084514180325849070552122171924176917642001416635954401589378633568231470947976
binomial $\operatorname{Binom}(n,p)$
13, 1/3
bits:
2.805413007599093431670431630028159747835832159113235455035417998382400515029657444373386452851581045
binomial $\operatorname{Binom}(n,p)$
13, 1/4
nats:
1.851680671530137282023410308032413709292143787394824874559081524775212452266064072733033668156216679
binomial $\operatorname{Binom}(n,p)$
13, 1/4
bits:
2.671410522126474626400199531706340176230473786395871396952846644295148923934443322788040254013259026
binomial $\operatorname{Binom}(n,p)$
13, 1/10
nats:
1.425334317538911280471013833900516310823521037957303710637512457789162811219209125583111098650044819
binomial $\operatorname{Binom}(n,p)$
13, 1/10
bits:
2.056322751522242362882003948459146033711056622305512771762044044133964056409481845763110558009180826
binomial $\operatorname{Binom}(n,p)$
14, 1/2
nats:
2.044810607569574310097131987022595035381296932656343413460113955319074702427677818472436995414774400
binomial $\operatorname{Binom}(n,p)$
14, 1/2
bits:
2.950038123097773117199320809785227459133997815943419761231124115081723396005797455683095552738278515
binomial $\operatorname{Binom}(n,p)$
14, 1/3
nats:
1.982127051772631267038733603341220540414263206555848717639016629673747599504872062318668308121476102
binomial $\operatorname{Binom}(n,p)$
14, 1/3
bits:
2.859604868004236754469776584737670976291943473787932461472627890195730103528486426623993444107023148
binomial $\operatorname{Binom}(n,p)$
14, 1/4
nats:
1.890055593667081970476169651086092613558982076207029346128194209473706941914693162584327345882057333
binomial $\operatorname{Binom}(n,p)$
14, 1/4
bits:
2.726773831987944830725370862669671955253380132269404874652388584120976338497651020351738741439709388
binomial $\operatorname{Binom}(n,p)$
14, 1/10
nats:
1.467148496023943873448593335360874340216860559661202785474000903807950865698557665979142347740933686
binomial $\operatorname{Binom}(n,p)$
14, 1/10
bits:
2.116647859461444873591107353633652205046697234960516762738876580253625639947984603402871752715690359
binomial $\operatorname{Binom}(n,p)$
15, 1/2
nats:
2.079379967245641079165414630685158591940495514012834922444118236063811903470644124639919916748552804
binomial $\operatorname{Binom}(n,p)$
15, 1/2
bits:
2.999911166869141547410844698259566095586813102364131577409365213699961280416555189471445623651283084
binomial $\operatorname{Binom}(n,p)$
15, 1/3
nats:
2.017036179625133258761447271577200506491712295979448499982161637636627723088885626793584769199021101
binomial $\operatorname{Binom}(n,p)$
15, 1/3
bits:
2.909968093638800166761980069984400211981775606796396346171911535688404353602040669985071358389077182
binomial $\operatorname{Binom}(n,p)$
15, 1/4
nats:
1.925644313405758588822798446710057733093716389538935838120488855385162320497285236750530028335199396
binomial $\operatorname{Binom}(n,p)$
15, 1/4
bits:
2.778117501466520753740496601753406903430488237682881181251838058114618717334090389754011687505737723
binomial $\operatorname{Binom}(n,p)$
15, 1/10
nats:
1.506050493181588700357782924076941392319895512462625327165283464343927779588728314295235229483377822
binomial $\operatorname{Binom}(n,p)$
15, 1/10
bits:
2.172771577841455615401019754036382394778514585584508865771872561141571960643655921617190880454956768
binomial $\operatorname{Binom}(n,p)$
16, 1/2
nats:
2.111707237558766462059791986807525681437167382373984805166030901962917978572953483389504385970807223
binomial $\operatorname{Binom}(n,p)$
16, 1/2
bits:
3.046549559435364544578865033476697535494411851946415646661192579458723978332924687677702189318214814
binomial $\operatorname{Binom}(n,p)$
16, 1/3
nats:
2.049645691866555153783607320938430377950306813891715960870414527283978870942944137847467885732552658
binomial $\operatorname{Binom}(n,p)$
16, 1/3
bits:
2.957013675235307480372330645898418526045217670863791388735496255539816898038024052238917321607210808
binomial $\operatorname{Binom}(n,p)$
16, 1/4
nats:
1.958825853143070064467271159875180738062358981480849632420770970833024230602808771087457655761417193
binomial $\operatorname{Binom}(n,p)$
16, 1/4
bits:
2.825988344294600097214187337966283988204632578385470003423848941772561336401310126592304925408788228
binomial $\operatorname{Binom}(n,p)$
16, 1/10
nats:
1.542380641779777572018154504677993486041871252551152945643292856548860507977666793546346132900974247
binomial $\operatorname{Binom}(n,p)$
16, 1/10
bits:
2.225184903058821826216372768452124604793897925529678045674517416852147080759073635968683544745796152
binomial $\operatorname{Binom}(n,p)$
17, 1/2
nats:
2.142066530325249174617291687830349785979333504130304319077566546018774083162426432947215231842974671
binomial $\operatorname{Binom}(n,p)$
17, 1/2
bits:
3.090348760554465332574045553269944757295930225174113679038468876216670431307678348594015054430831425
binomial $\operatorname{Binom}(n,p)$
17, 1/3
nats:
2.080242829310522743310078917509452382916249236925365975101355818313948243667067264819786858514580974
binomial $\operatorname{Binom}(n,p)$
17, 1/3
bits:
3.001156013691117535260576531889433771540503604359976356509239810317407767375037218211634610726581333
binomial $\operatorname{Binom}(n,p)$
17, 1/4
nats:
1.989907672948883093653652768154109917629722539942970137279268115757963820559705245898996885621687110
binomial $\operatorname{Binom}(n,p)$
17, 1/4
bits:
2.870829931590250918030278062686599832125608554054968030938623758095748023318583079207722680814809374
binomial $\operatorname{Binom}(n,p)$
17, 1/10
nats:
1.576426450807346730890788572653416305536107825366155743086850928125608773686559391565242934226330080
binomial $\operatorname{Binom}(n,p)$
17, 1/10
bits:
2.274302622905948553378367093408530177574902440448449729105824597199499050886465271627594864875096358
binomial $\operatorname{Binom}(n,p)$
18, 1/2
nats:
2.170684369312037704678140431411315853507631963667937922350778279943722500086535067098495013706811253
binomial $\operatorname{Binom}(n,p)$
18, 1/2
bits:
3.131635574941663982078696160064133731513542973061895305641966178722482021647263575014206958952368244
binomial $\operatorname{Binom}(n,p)$
18, 1/3
nats:
2.109063762463817866757111455345178887374148964888516075259837294783182657564393302814035503089053049
binomial $\operatorname{Binom}(n,p)$
18, 1/3
bits:
3.042735831025168724453043928330984967764576287909762552910896012913782108827475846136043092430864647
binomial $\operatorname{Binom}(n,p)$
18, 1/4
nats:
2.019142243351521550461212400326495862325269580715468661394410802708049335124355849596686289743062591
binomial $\operatorname{Binom}(n,p)$
18, 1/4
bits:
2.913006501332656685897631666305193017424823527883154655377762153227574949442784919412622300203077247
binomial $\operatorname{Binom}(n,p)$
18, 1/10
nats:
1.608432756398207206950308126746680429323826431077520480175046630812473955497146535473313148754088853
binomial $\operatorname{Binom}(n,p)$
18, 1/10
bits:
2.320477961259059665996529035071245997653186804333746264626245860536318299127471355286244978220466233
binomial $\operatorname{Binom}(n,p)$
19, 1/2
nats:
2.197750160447903330211463800855270884725604101935983115893258022648254228604079301187936897321322294
binomial $\operatorname{Binom}(n,p)$
19, 1/2
bits:
3.170683257591113784053150292485944822693711318183845395158648384103295701683247472395632315510292690
binomial $\operatorname{Binom}(n,p)$
19, 1/3
nats:
2.136305054405391190264825348982171562692434447661947190835990225404521077153541291542956451294852026
binomial $\operatorname{Binom}(n,p)$
19, 1/3
bits:
3.082036707816685039779205644338042588001030369018941970369660121722540093694882436748177375745805662
binomial $\operatorname{Binom}(n,p)$
19, 1/4
nats:
2.046739195871583596483598754098660604185312464248675627170260455708059742177640008714625708201527896
binomial $\operatorname{Binom}(n,p)$
19, 1/4
bits:
2.952820487876998381131364604471839447130337948852399981874491013271162165829309370093380097919533524
binomial $\operatorname{Binom}(n,p)$
19, 1/10
nats:
1.638609542355849424047184301281468910599923447148408901065595702160905837717155888344549480768251747
binomial $\operatorname{Binom}(n,p)$
19, 1/10
bits:
2.364013860710117801164872244623699093337769290443405979520320787991693633508569158208675624181039703
binomial $\operatorname{Binom}(n,p)$
20, 1/2
nats:
2.223423915810262567239544677696427764524255001784966920674530494917648585734516288333786296496313194
binomial $\operatorname{Binom}(n,p)$
20, 1/2
bits:
3.207722657133385887058394919338853818057502457586999632608268480481497539630388631016398679455331662
binomial $\operatorname{Binom}(n,p)$
20, 1/3
nats:
2.162132088127826855342146759871157421967391186762613855327834613668725454243338782300492123243183177
binomial $\operatorname{Binom}(n,p)$
20, 1/3
bits:
3.119297241288914998405254548352603805258171437816086398419924694298172711294737962627160490456808272
binomial $\operatorname{Binom}(n,p)$
20, 1/4
nats:
2.072874365788884422878766277871792819603537415907228624388941131910824278702235934922448518078076433
binomial $\operatorname{Binom}(n,p)$
20, 1/4
bits:
2.990525567909478703261188214559997777502009048769751975121912005773330791313942086996857677547404871
binomial $\operatorname{Binom}(n,p)$
20, 1/10
nats:
1.667138051650713579803409935157578572182980761903829093350385429489717633412968525026779786858013876
binomial $\operatorname{Binom}(n,p)$
20, 1/10
bits:
2.405171799593773016977828137848264973610022741627892474692174321367237803634479624417841165835336386
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/2
nats:
1.386294361119890618834464242916353136151000268720510508241360018986787243939389431211726653992837375
comment: In bits this entropy is exactly $2$.
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/3
nats:
1.909542504884438455351271467851223977791471404747737846962722981925695635716437469409120610448358891
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/3
bits:
2.754887502163468544361216831843449526279443223077443181367257963623294683383075687566841424754264726
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/4
nats:
2.249340578475233401152121260897835430661529401413793677761357075034666096101730824226059351530153234
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/4
bits:
3.245112497836531455638783168156550473720556776922556818632742036376705316616924312433158575245735274
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/5
nats:
2.502012117690939397665939694465525653024005696460546576597799381423745962780731098303762774494548554
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/5
bits:
3.609640474436811739351597147446950879324156965122903060273781979079673883043126079250698716796850775
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/10
nats:
3.250829733914482395065500282238179392356184845478239629108781004200828818838563398634888464248805385
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/10
bits:
4.689955935892812212535893303833204600971654591781147032344016176419579665787798033100348885068113193
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/20
nats:
3.970304866917451128532950316981598306021826495097117050858913717442763293721195431268201349221281698
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/20
bits:
5.727939142319122575329519554557949486119984036922465964046158029891956576268671467751341837916049225
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/6
nats:
2.703367253197828131871067484153275438709937381755439625569006601414432552591533271726205100395006604
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/6
bits:
3.900134529890125349370836516239948173234729481031983302460733948166915483723025295882984132711678677
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/7
nats:
2.870814228018863130862605053818044471121648953975000084083484990777731548134662779636734999254196031
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/7
bits:
4.141709450076291663371351556935916607928299935608099123006526141245656575269199530351344144682946043
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/8
nats:
3.014161290051494290276101710893979526352410117573097779680589178278387888008188300027110357798947812
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/8
bits:
4.348515545596771247906214779377184339512813618237014514258957931507754057964649094514973005808524504
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/9
nats:
3.139488862587287019100843349609225049842826816163364032328177777633450350662288229184363738733129097
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/9
bits:
4.529325012980811266167300991060697157676659338464659088203547781739768100298454125401048548525588358
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/11
nats:
3.350997070841619144501464810772780222027760507023859167070965790760583766272079041465629737132999722
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/11
bits:
4.834466856136646339489799219089820787087232852253642663893254144830289050833949721303006858562203825
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/12
nats:
3.442031796673927738075134400450123791543114525207530592308407431055068582850376933198507117649850861
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/12
bits:
4.965802203643604359251873813390075559382226109810323941028213751503541916612100870462961406861153529
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/13
nats:
3.525461849543973845938822982282595772886593363363572993734939407133662229321351555170361719047529788
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/13
bits:
5.086166327180323907713697177963233432051900330973713078530993981463938692697026151648529194637954747
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/14
nats:
3.602460967613643034620851368269846305507704813306937682673390593408588110887363435314834933453699649
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/14
bits:
5.197252572972259419029005935908599783804699540242485167482058301027374457841372909054159094727937340
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/15
nats:
3.673950401919530386623880443611711994620190586179377408589151144641470427789305205046938935423612490
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/15
bits:
5.300390025321320435673305160312468948395314247229954116175551618371003181974054864616249285778924966
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/16
nats:
3.740666539303348812762787221092600194235629917687328659013387230902092592166460826219008434167120509
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/16
bits:
5.396641065872222060139124398441899730630312989244074912342364244644504933955243324413696725838124042
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/17
nats:
3.803207293119173561539232730519850748167442614901325120329520936421781982555369976730359828866754105
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/17
bits:
5.486868301255769940319122183776874018191543799861951698152354094573840653805495161054833581342632645
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/18
nats:
3.862064793175291095496918211614952487669211286145913452568194637831905962708595912850094302592341464
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/18
bits:
5.571781724705852592015479798344520297161774877067366478254741468905695546791413089747263515708544070
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/19
nats:
3.917648962031404280432514490411113614841543464579872715600008309945681587895320797213302233269166448
geometric with success probability $p$ on $\{1,2,\ldots\}$
1/19
bits:
5.651972729466501849742267053108459715823325146639828100641874322947202755306924598449355932220177162
geometric with success probability $p$ on $\{1,2,\ldots\}$
2/3
nats:
0.9547712524422192276756357339256119888957357023738689234813614909628478178582187347045603052241794456
geometric with success probability $p$ on $\{1,2,\ldots\}$
2/3
bits:
1.377443751081734272180608415921724763139721611538721590683628981811647341691537843783420712377132363
geometric with success probability $p$ on $\{1,2,\ldots\}$
3/4
nats:
0.7497801928250778003840404202992784768871764671379312259204523583448886987005769414086864505100510780
geometric with success probability $p$ on $\{1,2,\ldots\}$
3/4
bits:
1.081704165945510485212927722718850157906852258974185606210914012125568438872308104144386191748578425
geometric with success probability $p$ on $\{1,2,\ldots\}$
4/5
nats:
0.6255030294227348494164849236163814132560014241151366441494498453559364906951827745759406936236371384
geometric with success probability $p$ on $\{1,2,\ldots\}$
4/5
bits:
0.9024101186092029348378992868617377198310392412807257650684454947699184707607815198126746791992126939
geometric with success probability $p$ on $\{1,2,\ldots\}$
5/6
nats:
0.5406734506395656263742134968306550877419874763510879251138013202828865105183066543452410200790013208
geometric with success probability $p$ on $\{1,2,\ldots\}$
5/6
bits:
0.7800269059780250698741673032479896346469458962063966604921467896333830967446050591765968265423357354
geometric with success probability $p$ on $\{1,2,\ldots\}$
6/7
nats:
0.4784690380031438551437675089696740785202748256625000140139141651296219246891104632727891665423660052
geometric with success probability $p$ on $\{1,2,\ldots\}$
6/7
bits:
0.6902849083460486105618919261559861013213833226013498538344210235409427625448665883918906907804910072
geometric with success probability $p$ on $\{1,2,\ldots\}$
7/8
nats:
0.4305944700073563271823002444134256466217728739390139685257984540397696982868840428610157653998496874
geometric with success probability $p$ on $\{1,2,\ldots\}$
7/8
bits:
0.6212165065138244639866021113395977627875448026052877877512797045011077225663784420735675722583606435
geometric with success probability $p$ on $\{1,2,\ldots\}$
8/9
nats:
0.3924361078234108773876054187011531312303533520204205040410222222041812938327860286480454673416411371
geometric with success probability $p$ on $\{1,2,\ldots\}$
8/9
bits:
0.5661656266226014082709126238825871447095824173080823860254434727174710125373067656751310685656985447
geometric with success probability $p$ on $\{1,2,\ldots\}$
9/10
nats:
0.3612033037682758216739444758042421547062427606086932921231978893556476465376181554038764960276450428
geometric with success probability $p$ on $\{1,2,\ldots\}$
9/10
bits:
0.5211062150992013569484325893148005112190727324201274480382240196021755184208664481222609872297903548
geometric with success probability $p$ on $\{1,2,\ldots\}$
10/11
nats:
0.3350997070841619144501464810772780222027760507023859167070965790760583766272079041465629737132999722
geometric with success probability $p$ on $\{1,2,\ldots\}$
10/11
bits:
0.4834466856136646339489799219089820787087232852253642663893254144830289050833949721303006858562203825
geometric with success probability $p$ on $\{1,2,\ldots\}$
11/12
nats:
0.3129119815158116125522849454954657992311922295643209629371279482777335075318524484725915561499864419
geometric with success probability $p$ on $\{1,2,\ldots\}$
11/12
bits:
0.4514365639676003962956248921263705053983841918009385400934739774094129015101909882239055824419230481
geometric with success probability $p$ on $\{1,2,\ldots\}$
12/13
nats:
0.2937884874619978204949019151902163144072161136136310828112449505944718524434459629308634765872941490
geometric with success probability $p$ on $\{1,2,\ldots\}$
12/13
bits:
0.4238471939316936589761414314969361193376583609144760898775828317886615577247521793040440995531628956
geometric with success probability $p$ on $\{1,2,\ldots\}$
13/14
nats:
0.2771123821241263872785270283284497158082849856389952063594915841083529316067202642549873025733615115
geometric with success probability $p$ on $\{1,2,\ldots\}$
13/14
bits:
0.3997886594594045706945389181468153679849768877109603974986198693097980352185671468503199303636874877
geometric with success probability $p$ on $\{1,2,\ldots\}$
14/15
nats:
0.2624250287085378847588486031151222853300136132985269577563679389029621734135218003604956382445437493
geometric with success probability $p$ on $\{1,2,\ldots\}$
14/15
bits:
0.3785992875229514596909503685937477820282367319449967225839679727407859415695753474725892346984946405
geometric with success probability $p$ on $\{1,2,\ldots\}$
15/16
nats:
0.2493777692868899208508524814061733462823753278458219106008924820601395061444307217479338956111413673
geometric with success probability $p$ on $\{1,2,\ldots\}$
15/16
bits:
0.3597760710581481373426082932294599820420208659496049941561576163096336622636828882942464483892082695
geometric with success probability $p$ on $\{1,2,\ldots\}$
16/17
nats:
0.2377004558199483475962020456574906717604651634313328200205950585263613739097106235456474893041721316
geometric with success probability $p$ on $\{1,2,\ldots\}$
16/17
bits:
0.3429292688284856212699451364860546261369714874913719811345221309108650408628434475659270988339145403
geometric with success probability $p$ on $\{1,2,\ldots\}$
17/18
nats:
0.2271802819514877114998187183302913228040712521262302030922467434018768213357997595794173119171965567
geometric with success probability $p$ on $\{1,2,\ldots\}$
17/18
bits:
0.3277518661591677995303223410790894292448102868863156751914553805238644439289066523380743244534437689
geometric with success probability $p$ on $\{1,2,\ldots\}$
18/19
nats:
0.2176471645573002378018063605783952008245301924766595953111115727747600882164067109562945685149536916
geometric with success probability $p$ on $\{1,2,\ldots\}$
18/19
bits:
0.3139984849703612138745703918393588731012958414799904500356596846081779308503846999138531073455653979
geometric with success probability $p$ on $\{1,2,\ldots\}$
19/20
nats:
0.2089634140482869015017342272095578055800961313209008974136270377601454365116418648035895446958569314
geometric with success probability $p$ on $\{1,2,\ldots\}$
19/20
bits:
0.3014704811746906618594483976083131308484202124696034717919030542048398198036142877763864125218973276
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
2, 1/2
nats:
1.879446901369988737925788796157318555100512433922042600578113052861162305480701611006017410718057259
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
2, 1/2
bits:
2.711468724220611478455312866757461691832922457629908654148941494435117824326743107673891892166375716
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
2, 1/3
nats:
2.449867104579724689101935770018060700850781293299710434708526875716573623656414507411149302607820754
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
2, 1/3
bits:
3.534411122614172302399089788662596008978492107551128562669327673613488531954707255572644391095372123
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
2, 1/4
nats:
2.805417974802155386714640721728335314455627537410893676914064465364988721563584019301420108927106471
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
2, 1/4
bits:
4.047362599867828479671397437794379190413741192229037728235451854040454717915132847439611885682872696
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
2, 1/10
nats:
3.824231985038146674850500425972198754184672624619114773187337091564020102471937812117569704276518706
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
2, 1/10
bits:
5.517200520023490714553629076400888082127447847765227725887394437281408174615990746919639767794903455
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
3, 1/2
nats:
2.159602485630811620012601119650848258667065315834938900741333872083865742214323643137821920133446330
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
3, 1/2
bits:
3.115647796311050779535873450451879809622313225025739365623644668625130642093557685747070761375193152
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
3, 1/3
nats:
2.730098044633693543146062168051642813179599074191337631500254290553572546766619188911606339561276595
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
3, 1/3
bits:
3.938698910133685551761085832307448105028119671101382761408863728879034105161536477177912234858875932
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
3, 1/4
nats:
3.083097240951922637197144180195810670198606399721134551957041829215929410504898952918999292105844901
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
3, 1/4
bits:
4.447969100099784295686643272643995038981149913619978091953091028354179149934313319379988740428386180
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
3, 1/10
nats:
4.096539445838805839523459194428109521218538623476097589951459824280402106997762802472410186292422924
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
3, 1/10
bits:
5.910057143317667488360733008613298969559669408420038695701362807682073655530623029481239210390949538
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
4, 1/2
nats:
2.346363347587143987539751892729112036624632263462642478634989332204783342265270592599639057599919211
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
4, 1/2
bits:
3.385086765687599754964928686428639950660618372377804012243372175484322094394964718989275094401232184
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
4, 1/3
nats:
2.911513097736697861019405938318775954212687596456503724817299722978323273439689272213272012966385246
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
4, 1/3
bits:
4.200425507587997834008370633356179386578991947856242305961671347912645883429980684748564575466487796
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
4, 1/4
nats:
3.262086579310003689959415571992778032064618586465496322993423890925318072542832233048726092693040789
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
4, 1/4
bits:
4.706196130920984546532648707880344135611277419271709713910260375029858122531974200756423899248507727
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
4, 1/10
nats:
4.272837953295725805724313083039532558496559058474614057493275072199571602767671617094046043666500600
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
4, 1/10
bits:
6.164402125741891859177045220415489012993207890237866142348135519904411128973273457465634981136538946
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
5, 1/2
nats:
2.483619150583802605562173905513994399670192904332732875099860366839411599285545380546957675711634023
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
5, 1/2
bits:
3.583105032004111666070963738862561101686470002006546674905011373428116160649116775476381357539647803
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
5, 1/3
nats:
3.044375976937700299109851396698909384452670977918133547065177045252981774097929905960529939050059743
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
5, 1/3
bits:
4.392106124529513452275536562145327021989956085007890092699036251185265745372166310442733511973920097
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
5, 1/4
nats:
3.393592382478125184568360171607053703134349869249672294010630521083591017269045755383931060565907164
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
5, 1/4
bits:
4.895918900999753559794296683271328828771970765001241167663230603895475643094670293068877903314421277
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
5, 1/10
nats:
4.403176828671754228578305727576861159781646038679826337222800037026905826897151266270601933316501943
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
5, 1/10
bits:
6.352441374881932690493257863936960449965086412926850424968971571687067591769433371396454255963040627
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
6, 1/2
nats:
2.591291695951424195371350623153102952307365231053966774394307045355189284354839470362210447109646226
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
6, 1/2
bits:
3.738443679245871263094694498391608695068929465023119394473423814144673706304638435421628761800973870
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
6, 1/3
nats:
3.149085770859069736133955193010718286920116800760524741613002236811312536129417582341442892053557691
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
6, 1/3
bits:
4.543170424952378464266099573547086029922628767699342378594929576539360368535108293680700451404071433
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
6, 1/4
nats:
3.497565413071357586776615912664227567575294152321964867542326420592651734354784048226522660755759414
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
6, 1/4
bits:
5.045920276622806423577129855224952410902265918418637868810974302361872666294011334140358178947896431
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
6, 1/10
nats:
4.506575884046698504020046617684940330183166132751328752398296056598671644031170993073252399665853757
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
6, 1/10
bits:
6.501614679303968113533359252614631543666064412182277714875481114022307740191274032718326488264251596
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
7, 1/2
nats:
2.679668528945075960943083140404993073002061525033255817421514110721922765458717757689812249592629283
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
7, 1/2
bits:
3.865944497935284787361991451555035252110944017933382779306912505447064676374867209754016500579699738
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
7, 1/3
nats:
3.235529793580250893536859058587691996264906378865365704046175513946143211890424686054166035138065781
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
7, 1/3
bits:
4.667882787846719395976848212063342874633686453681245648202490340953250000767180855513199705020552695
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
7, 1/4
nats:
3.583582543152124785753146295468219978536449249496344751414840284065868450574188126309736324810495951
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
7, 1/4
bits:
5.170016763621830141983183997386615311088067034292921461156027373144744235381743384988088704263124433
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
7, 1/10
nats:
4.592264428308361232133418250572350637019450815640842544996433250574679842373959487904250538182735884
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
7, 1/10
bits:
6.625237117171263373813948485285472273259823247244383130908027048066716246718626416397629415874875444
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
8, 1/2
nats:
2.754588631980323463205976651105495546790927827281703094629101148286790838358658492693927024191928246
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
8, 1/2
bits:
3.974031359047126534123026950376289469643360963045388132684340527005615789046151076936042622800236399
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
8, 1/3
nats:
3.309168431412236876319500797284692877809412583333243660739313286505193885565360347218120097800379833
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
8, 1/3
bits:
4.774120885464743981087356562640274070678265685955748297779692707724663715415949627289268445077347594
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
8, 1/4
nats:
3.656951445667729262850747948648301419117847583415276396174313381087052256049068916813730275744625302
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
8, 1/4
bits:
5.275865715436558513268217403783844332338630503472058785754601672851856275495216449273385438475914913
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
8, 1/10
nats:
4.665421277814408941696014684628752712365488801277787455346415162704871219521067007584441335878993395
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
8, 1/10
bits:
6.730780141160698616611880324939210485640152621350730243767971172110116541585501401427538937871602923
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
9, 1/2
nats:
2.819624074503889115839126687935684535990942074836195198494923473083851328800023206420327677550394053
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
9, 1/2
bits:
4.067857669457893912876144909344075283008976027618465587599758205854999141655900020044966231105736001
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
9, 1/3
nats:
3.373323834047520856084623522341011748335782853633496765441233330993080000289264146428976721398405721
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
9, 1/3
bits:
4.866677566692902913013169420962884119884071767179663599843283630143199448511560274488109355225273307
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
9, 1/4
nats:
3.720922351794979839946667254461993478464442545271333759307270753068916484315722201006571445163548637
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
9, 1/4
bits:
5.368156224467516324364420347060658273452577784510998206696202208414721618492330165888221263506010427
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
9, 1/10
nats:
4.729243471379188572086396087517198087965729499739399490243101528435308618002083080562252804923578886
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
9, 1/10
bits:
6.822856103315261702725647996690290815935359435862464307846333694509709045795941519733308448374570932
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
10, 1/2
nats:
2.877098210794647635265328380692670580739269121320515312631039129165275179695762828767124516344119412
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
10, 1/2
bits:
4.150775320863947630722171133737885449295870885246950094832225370424433250907809552109454928613195122
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
10, 1/3
nats:
3.430168279359647660797625165510119006740096483615321154332986778280490677674774846616001543424642280
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
10, 1/3
bits:
4.948686766046792137817818656654195383840529638592860536611923069072070882970474994817410560505475502
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
10, 1/4
nats:
3.777632439167529815198128832458566399337494675942545817055974920014516285611858052542849012667986687
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
10, 1/4
bits:
5.449971586288273998314207782771346610777598428328893835458247101761065774695882232563150202289125783
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
10, 1/10
nats:
4.785843152910122319598680897013319929343095933543888146045893710117799204336282555136944715950496492
negative binomial $\operatorname{NB}(r,p)$, failures before the $r$th success with success probability $p$
10, 1/10
bits:
6.904512183175834472579459092817319189411425758731400099897986900147206831659834870993656702635258307
Poisson$(\lambda)$
1/2
nats:
0.9276374674957973741380689368997581161181488664823499650950432870081292570454866484877211846538914968
Poisson$(\lambda)$
1/2
bits:
1.338297974098983856413811741912613418168564396089567771516892892268879012163659973754392647533261072
Poisson$(\lambda)$
1
nats:
1.304842242256251484308800012107587605872065984275157220597445437553228920185141353023669020482477252
Poisson$(\lambda)$
1
bits:
1.882489432045529431148225220161389460789969828965417543855549868681617086658698595848528595674821755
Poisson$(\lambda)$
3/2
nats:
1.539404652817375537420505972498112187463117204944891761208332163402378303606451029641360422929993834
Poisson$(\lambda)$
3/2
bits:
2.220891458541024119128154395955314249913987338184135433427170610929808291199388515682229815447931185
Poisson$(\lambda)$
2
nats:
1.704882643932983838377586644492340397420976065766115801126708886596714230612459356368362295216890990
Poisson$(\lambda)$
2
bits:
2.459625735699780160326968433582661250941312804283989504850529101691252519175780838299821850064095362
Poisson$(\lambda)$
5/2
nats:
1.830726607926971682605599957334357820035222633779063185594791050580240543441090320215311599127137182
Poisson$(\lambda)$
5/2
bits:
2.641180198479715692042790979191877764391702715212450163151039267763513335847241238169855842712652037
Poisson$(\lambda)$
3
nats:
1.931470198148568933920429874127834053141403525336052813746881945019538729813054702547951673861818325
Poisson$(\lambda)$
3
bits:
2.786522476493763912625392660677598793417648784264444082499603201777175474140335917082785455885589506
Poisson$(\lambda)$
7/2
nats:
2.015172522512972281191703257567668482151311594224260979686239537574506009959929236249385425162933419
Poisson$(\lambda)$
7/2
bits:
2.907279404765168077763981946958716181565140299731930334643135942293590084967320392355186793891106727
Poisson$(\lambda)$
4
nats:
2.086672699880963843364905413809789785036141186912488866865116363393855498254560539498013526411849621
Poisson$(\lambda)$
4
bits:
3.010432356076650800573001069074024808015655368347941892641591307264329311555754173685781972285967802
Poisson$(\lambda)$
9/2
nats:
2.149057657291177720662766283583869089488940127886608659360632153135190596179776291374245081653255474
Poisson$(\lambda)$
9/2
bits:
3.100434824758435550997181059363749150563688468453148571200660433993762837860831402693911623626509882
Poisson$(\lambda)$
5
nats:
2.204395243428367907085868718265451070467333943465290792661037432762168690515573557212543455905145669
Poisson$(\lambda)$
5
bits:
3.180270085853325681614056203858280644121621748014434509458697795052746386504081507283025505277318451
Poisson$(\lambda)$
11/2
nats:
2.254128397205339932352588666478317114386028847864020174964231291683236504300151061310609361205464332
Poisson$(\lambda)$
11/2
bits:
3.252019860175131442525439909447227502047372674140368057392736866676977844171176147424120252998512405
Poisson$(\lambda)$
6
nats:
2.299299563162719628833883849800045225698348435423323454727939858103141962017085434319194087898010972
Poisson$(\lambda)$
6
bits:
3.317188077293015495639602812475706867310217218037655142815698360509239313845274138726121649581034856
Poisson$(\lambda)$
13/2
nats:
2.340684595915005417759062959845431555694587255351110902445735483368890061994314886984808320522044646
Poisson$(\lambda)$
13/2
bits:
3.376894058811765531713619702322029310999870731360610646323948074147436043736747185373883447486102309
Poisson$(\lambda)$
7
nats:
2.378875915025292790756490494142431101060468544485704306409238317403771896450309845448426578662224143
Poisson$(\lambda)$
7
bits:
3.431992485497185022879489609008425074874441390853514411247465825092867379581380835445612747992943408
Poisson$(\lambda)$
15/2
nats:
2.414335980371920956848660018534165662085899103747658224941220722760423680573272963984444957567045024
Poisson$(\lambda)$
15/2
bits:
3.483150545922364059340702927887187831910386129631038879933892121175569900975006475066036640978476130
Poisson$(\lambda)$
8
nats:
2.447432766544303521145548835547853824202579976376958155903602052273557668064215027567353141305212156
Poisson$(\lambda)$
8
bits:
3.530899115202622801642769555494344826907863480167108226115693723473248554820963499654229659457133342
Poisson$(\lambda)$
17/2
nats:
2.478463961751760384572910585812540462630704011338499817479853070593015375121234571232483361502084268
Poisson$(\lambda)$
17/2
bits:
3.575667666641278186549453571040999481535369738125960098441341029587031497697636499449120556096570763
Poisson$(\lambda)$
9
nats:
2.507673896080288983307899256236729597427714249478538371667415759186631878385724151976413858940196683
Poisson$(\lambda)$
9
bits:
3.617808694041738689192020417041845135204635804161951197483586803302070332816148388379208187632523594
Poisson$(\lambda)$
19/2
nats:
2.535265656177975599923122887122388088267447942333826643597252635247341844025305790685167167348694253
Poisson$(\lambda)$
19/2
bits:
3.657615189504069151528682240787388045777181971586049629857788124057767858975400124538181277122040463
Poisson$(\lambda)$
10
nats:
2.561409935274909122596534696011572878698544369530912100000667004818623769766638851673775509467973981
Poisson$(\lambda)$
10
bits:
3.695333411304832131328319876231498997315132843521031068595099901617290103726027857147682893950738059
Poisson$(\lambda)$
11
nats:
2.609914772529964161676277949405298375717912894829116304709278349102478813419169065582539960048254826
Poisson$(\lambda)$
11
bits:
3.765311099471826276535363975965011024604823913841807286257761630206720976366213409368797014795449274
Poisson$(\lambda)$
12
nats:
2.654119759193953918448335987878548541661362776799334656918587763835290057395884361905779329182442472
Poisson$(\lambda)$
12
bits:
3.829085414514527060808338882779417657276695424440188940110712685238850240503430676567908107201734572
Poisson$(\lambda)$
13
nats:
2.694726988440934805110961128293260498543857305929112398328495603968114266330653426695040349923164309
Poisson$(\lambda)$
13
bits:
3.887669262773387661846152043739604545190072991584634004461327060852813301161396284635288477828148974
Poisson$(\lambda)$
14
nats:
2.732279117514093289816313994292547256438650865354550028685514005388522950003731839614356520148584870
Poisson$(\lambda)$
14
bits:
3.941845533162055673478975141858786516297399989561695703619207833149758297773820733347130039947790525
Poisson$(\lambda)$
15
nats:
2.767204386781281645709122804666784127619841120604633162307118331245380942090578191697760008076242512
Poisson$(\lambda)$
15
bits:
3.992232045935540035642007804819096397176270482210215783708782093486466719098225811644873570063013614
Poisson$(\lambda)$
16
nats:
2.799846746366404883161529115997215534963651806403194667798276856182137929455106883845833280757558198
Poisson$(\lambda)$
16
bits:
4.039325016231911651304395676781591241896018381229093137751574699315737633201032297768672627042994624
Poisson$(\lambda)$
17
nats:
2.830486654592801591065821310765284142091578493426671526871562930910987791084869048303779491944531987
Poisson$(\lambda)$
17
bits:
4.083529059883427136088379824870694276952723411084483570969287226846491699267734110897036010382572312
Poisson$(\lambda)$
18
nats:
2.859355821508481381140110271800580190314589983190394980296541927767915649013136900939404306910534449
Poisson$(\lambda)$
18
bits:
4.125178463827274100445802854081470275331922590989326931749847361269307494498906843958151418188165731
Poisson$(\lambda)$
19
nats:
2.886647901001217788964592056359279022509628800407729944607314757041114728866668175279991185604526433
Poisson$(\lambda)$
19
bits:
4.164552611566992292021400797092547888372610023933355541736114425840975872630037189065582161043429559
Poisson$(\lambda)$
20
nats:
2.912526400182318058528039532827949715257857927145863882972145255583603459312485415107806276842359088
Poisson$(\lambda)$
20
bits:
4.201887394001214751472641352896721724969469810372668779240524121494806876950183971882121389614160943
uniform on $\{1,\ldots,n\}$
2
nats:
0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
comment: In bits this entropy is exactly $1$.
uniform on $\{1,\ldots,n\}$
3
nats:
1.098612288668109691395245236922525704647490557822749451734694333637494293218608966873615754813732089
uniform on $\{1,\ldots,n\}$
3
bits:
1.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242
uniform on $\{1,\ldots,n\}$
4
nats:
1.386294361119890618834464242916353136151000268720510508241360018986787243939389431211726653992837375
comment: In bits this entropy is exactly $2$.
uniform on $\{1,\ldots,n\}$
5
nats:
1.609437912434100374600759333226187639525601354268517721912647891474178987707657764630133878093179611
uniform on $\{1,\ldots,n\}$
5
bits:
2.321928094887362347870319429489390175864831393024580612054756395815934776608625215850139743359370155
uniform on $\{1,\ldots,n\}$
6
nats:
1.791759469228055000812477358380702272722990692183004705855374343130887915188303682479479081810150776
uniform on $\{1,\ldots,n\}$
6
bits:
2.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242
uniform on $\{1,\ldots,n\}$
7
nats:
1.945910149055313305105352743443179729637084729581861188459390149937579862752069267787658498587871527
uniform on $\{1,\ldots,n\}$
7
bits:
2.807354922057604107441969317231830808641026625966140783677291724070320848862192986497860999170210785
uniform on $\{1,\ldots,n\}$
8
nats:
2.079441541679835928251696364374529704226500403080765762362040028480180865909084146817589980989256063
comment: In bits this entropy is exactly $3$.
uniform on $\{1,\ldots,n\}$
9
nats:
2.197224577336219382790490473845051409294981115645498903469388667274988586437217933747231509627464178
uniform on $\{1,\ldots,n\}$
9
bits:
3.169925001442312362907477887895633017519628815384962120911505309082196455588717125044560949836176484
uniform on $\{1,\ldots,n\}$
10
nats:
2.302585092994045684017991454684364207601101488628772976033327900967572609677352480235997205089598298
uniform on $\{1,\ldots,n\}$
10
bits:
3.321928094887362347870319429489390175864831393024580612054756395815934776608625215850139743359370155
uniform on $\{1,\ldots,n\}$
11
nats:
2.397895272798370544061943577965129299821706853937417175218567709130573623913236713075054708002634791
uniform on $\{1,\ldots,n\}$
11
bits:
3.459431618637297256199363046725792958703231525681768071312801645726330619720018352709491299286900489
uniform on $\{1,\ldots,n\}$
12
nats:
2.484906649788000310229709479838878840798490826543259959976054352624281537157998398085342408806569464
uniform on $\{1,\ldots,n\}$
12
bits:
3.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242
uniform on $\{1,\ldots,n\}$
13
nats:
2.564949357461536736053487441565318604805267944760207116419045510663464667324410179399574663440489489
uniform on $\{1,\ldots,n\}$
13
bits:
3.700439718141092160396812654256694733628436401791037369538463525842855186633025300147376530281154896
uniform on $\{1,\ldots,n\}$
14
nats:
2.639057329615258614522584864901356297712584863942116442580070159430973484721763983393521825584290215
uniform on $\{1,\ldots,n\}$
14
bits:
3.807354922057604107441969317231830808641026625966140783677291724070320848862192986497860999170210785
uniform on $\{1,\ldots,n\}$
15
nats:
2.708050201102210065996004570148713344173091912091267173647342225111673280926266731503749632906911700
uniform on $\{1,\ldots,n\}$
15
bits:
3.906890595608518529324058373437206684624645800717061672510509050357033004402983778372420218277458397
uniform on $\{1,\ldots,n\}$
16
nats:
2.772588722239781237668928485832706272302000537441021016482720037973574487878778862423453307985674750
comment: In bits this entropy is exactly $4$.
uniform on $\{1,\ldots,n\}$
17
nats:
2.833213344056216080249534617873126535588203012585744787297237737882292575800931280912094868037502948
uniform on $\{1,\ldots,n\}$
17
bits:
4.087462841250339408254066010810404354011267282344820688126609064386696509047382068297343151843684273
uniform on $\{1,\ldots,n\}$
18
nats:
2.890371757896164692207722595303227977370481250005754157590068676768382208406912649353094836623882865
uniform on $\{1,\ldots,n\}$
18
bits:
4.169925001442312362907477887895633017519628815384962120911505309082196455588717125044560949836176484
uniform on $\{1,\ldots,n\}$
19
nats:
2.944438979166440460009027431887853537237379261299128818537960236409292702064197288714158383815739896
uniform on $\{1,\ldots,n\}$
19
bits:
4.247927513443585493793519422906834422693507569661534014581524730864565208205464886802708054172176520
uniform on $\{1,\ldots,n\}$
20
nats:
2.995732273553990993435223576142540775676601622989028230154007910460966231647047195841860532086016986
uniform on $\{1,\ldots,n\}$
20
bits:
4.321928094887362347870319429489390175864831393024580612054756395815934776608625215850139743359370155
uniform on $\{1,\ldots,n\}$
21
nats:
3.044522437723422996500597980365705434284575287404610640194084483575074155970678234661274253401603616
uniform on $\{1,\ldots,n\}$
21
bits:
4.392317422778760288895708261179647317400841033658621844133044378611419076656551549020141474088299027
uniform on $\{1,\ldots,n\}$
22
nats:
3.091042453358315853479175699423305867897206988297672429339247718623967245882931428680918034999053479
uniform on $\{1,\ldots,n\}$
22
bits:
4.459431618637297256199363046725792958703231525681768071312801645726330619720018352709491299286900489
uniform on $\{1,\ldots,n\}$
23
nats:
3.135494215929149690806752831810196118442380314840435741998635377482993245984798298198401092152994814
uniform on $\{1,\ldots,n\}$
23
bits:
4.523561956057012872294148244162668844498825125442555059494443732014778145562764696110754525862088214
uniform on $\{1,\ldots,n\}$
24
nats:
3.178053830347945619646941601297055408873990960903515214096734362117675159127693113691205735802988151
uniform on $\{1,\ldots,n\}$
24
bits:
4.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242
uniform on $\{1,\ldots,n\}$
25
nats:
3.218875824868200749201518666452375279051202708537035443825295782948357975415315529260267756186359222
uniform on $\{1,\ldots,n\}$
25
bits:
4.643856189774724695740638858978780351729662786049161224109512791631869553217250431700279486718740310
uniform on $\{1,\ldots,n\}$
26
nats:
3.258096538021482045470719563023495172880768079120462370539725520156858289294104895005437990436908176
uniform on $\{1,\ldots,n\}$
26
bits:
4.700439718141092160396812654256694733628436401791037369538463525842855186633025300147376530281154896
uniform on $\{1,\ldots,n\}$
27
nats:
3.295836866004329074185735710767577113942471673468248355204083000912482879655826900620847264441196266
uniform on $\{1,\ldots,n\}$
27
bits:
4.754887502163468544361216831843449526279443223077443181367257963623294683383075687566841424754264726
uniform on $\{1,\ldots,n\}$
28
nats:
3.332204510175203923939816986359532865788084998302371696700750168924367106691458698999385152580708902
uniform on $\{1,\ldots,n\}$
28
bits:
4.807354922057604107441969317231830808641026625966140783677291724070320848862192986497860999170210785
uniform on $\{1,\ldots,n\}$
29
nats:
3.367295829986474027183272032361911605494512913922744078921670351642780781137852333293367114817856423
uniform on $\{1,\ldots,n\}$
29
bits:
4.857980995127572120719773324627984762476860500075937048051943442788096336378103434971079945416980185
uniform on $\{1,\ldots,n\}$
30
nats:
3.401197381662155375413236691606889912248592046451522427768022234605066902895961447109612959903330387
uniform on $\{1,\ldots,n\}$
30
bits:
4.906890595608518529324058373437206684624645800717061672510509050357033004402983778372420218277458397
hypergeometric with population $N$, successes $K$, and draws $n$
10, 2, 2
nats:
0.7474814710871595424353002205959008132895071687690667637966026623611793342910689123605201166088516715
hypergeometric with population $N$, successes $K$, and draws $n$
10, 2, 2
bits:
1.078387811493832155036127519996328468007821418919499578678169076587709370683088927073809404822971040
hypergeometric with population $N$, successes $K$, and draws $n$
10, 2, 5
nats:
0.9950269901795212593824099014046464681916469702250978339087265011256030946130993020167734320217338660
hypergeometric with population $N$, successes $K$, and draws $n$
10, 2, 5
bits:
1.435520504282666614090633760401527364261389152593528447547751755851121579695036449572261092414304176
hypergeometric with population $N$, successes $K$, and draws $n$
10, 2, 8
nats:
0.7474814710871595424353002205959008132895071687690667637966026623611793342910689123605201166088516715
hypergeometric with population $N$, successes $K$, and draws $n$
10, 2, 8
bits:
1.078387811493832155036127519996328468007821418919499578678169076587709370683088927073809404822971040
hypergeometric with population $N$, successes $K$, and draws $n$
10, 5, 2
nats:
0.9950269901795212593824099014046464681916469702250978339087265011256030946130993020167734320217338660
hypergeometric with population $N$, successes $K$, and draws $n$
10, 5, 2
bits:
1.435520504282666614090633760401527364261389152593528447547751755851121579695036449572261092414304176
hypergeometric with population $N$, successes $K$, and draws $n$
10, 5, 5
nats:
1.235866276237501437415733987203138056856872737446438208882059671380121793756824376566409242187707681
hypergeometric with population $N$, successes $K$, and draws $n$
10, 5, 5
bits:
1.782978147929753081717861035505657921666942359953125499718248628755821319116336270569922458213986168
hypergeometric with population $N$, successes $K$, and draws $n$
10, 5, 8
nats:
0.9950269901795212593824099014046464681916469702250978339087265011256030946130993020167734320217338660
hypergeometric with population $N$, successes $K$, and draws $n$
10, 5, 8
bits:
1.435520504282666614090633760401527364261389152593528447547751755851121579695036449572261092414304176
hypergeometric with population $N$, successes $K$, and draws $n$
20, 2, 2
nats:
0.5172152102791041129553230505917346437690991633144458813666523863532445342889461111533164853628601624
hypergeometric with population $N$, successes $K$, and draws $n$
20, 2, 2
bits:
0.7461838189420059150723366394158745644797403294939282785539353884294489532850532969532924198940674680
hypergeometric with population $N$, successes $K$, and draws $n$
20, 2, 5
nats:
0.8496383026252035448134413853769020153464725539334277158947647840411483310544806008554235405745220360
hypergeometric with population $N$, successes $K$, and draws $n$
20, 2, 5
bits:
1.225768965746697493617973394319184582304366813935560756306483475438373216736192802460358732329646111
hypergeometric with population $N$, successes $K$, and draws $n$
20, 2, 10
nats:
1.020094373854497117875383236186510608448624328780448681460504608529814766433606037891406505153877138
hypergeometric with population $N$, successes $K$, and draws $n$
20, 2, 10
bits:
1.471685094398615244063493355225013427097456344992562161489360955080401215237848766597316160376424419
hypergeometric with population $N$, successes $K$, and draws $n$
20, 5, 2
nats:
0.8496383026252035448134413853769020153464725539334277158947647840411483310544806008554235405745220360
hypergeometric with population $N$, successes $K$, and draws $n$
20, 5, 2
bits:
1.225768965746697493617973394319184582304366813935560756306483475438373216736192802460358732329646111
hypergeometric with population $N$, successes $K$, and draws $n$
20, 5, 5
nats:
1.247673279616172846414580108421158201190032728727577037910456851035242831454395745575001619265833728
hypergeometric with population $N$, successes $K$, and draws $n$
20, 5, 5
bits:
1.800012053151921559224521579111895619063902103155378256424309736929313564440780981634547028902144749
hypergeometric with population $N$, successes $K$, and draws $n$
20, 5, 10
nats:
1.410178992589031228652819447306708741437597768533311273935985517608637596611516046058897830503149884
hypergeometric with population $N$, successes $K$, and draws $n$
20, 5, 10
bits:
2.034458239373989634221859674560713137552345239847121861610699015808455486701181968646063973142589464
hypergeometric with population $N$, successes $K$, and draws $n$
20, 10, 2
nats:
1.020094373854497117875383236186510608448624328780448681460504608529814766433606037891406505153877138
hypergeometric with population $N$, successes $K$, and draws $n$
20, 10, 2
bits:
1.471685094398615244063493355225013427097456344992562161489360955080401215237848766597316160376424419
hypergeometric with population $N$, successes $K$, and draws $n$
20, 10, 5
nats:
1.410178992589031228652819447306708741437597768533311273935985517608637596611516046058897830503149884
hypergeometric with population $N$, successes $K$, and draws $n$
20, 10, 5
bits:
2.034458239373989634221859674560713137552345239847121861610699015808455486701181968646063973142589464
hypergeometric with population $N$, successes $K$, and draws $n$
20, 10, 10
nats:
1.555951815188116695579963730964488847021215656874658798345139717291614830311643423806768088940376957
hypergeometric with population $N$, successes $K$, and draws $n$
20, 10, 10
bits:
2.244763967634076850971891622394606806258417704630886357073461114387040892888987441338589618719612136
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/2
nats:
0.8829244358028678611733744961944307354433879906766004350294190823427973390492067733167179077816810386
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/2
bits:
1.273790705012483395982892455866812915659499706530622071372273071913531835757766350845756697418032690
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/3
nats:
0.5957444426529752089282563657246481688718219798805927581642526774231175248251386656391035900374057623
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/3
bits:
0.8594775530526067848202872635752976425515522792528938072459351654796377067686578111829621668089720239
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/4
nats:
0.4616944452661272704618622200724521231693208414645724404058230192556863601350996894431734800131608207
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/4
bits:
0.6660842865914227603269938800400590287199881480149275216994252109417654285050799830833911066724897286
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/5
nats:
0.3815083617781486930406096444585622895676310367784354881304725093927229804815598207217323097148839578
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/5
bits:
0.5504002215950076730547759683670634669483144108551243294646871197679114329221134109077580761649880490
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/10
nats:
0.2146256733097977327700470387913745065872989396977681389797236452424608948005024455284672061437511875
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/10
bits:
0.3096393945314999423229881594359854230011559336076745152315600223435417085671692483856979602040546990
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/20
nats:
0.1215266341136756774640045442685343357925434586528077911320573803791387894668658499703513042745031811
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/20
bits:
0.1753258723717274267926351251017078948954670421256346383536396957327758234461123596411401495288593893
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/6
nats:
0.3273472620501924972867047929017136733273503482562817891295558531849540485355069257707215351221890187
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/6
bits:
0.4722622716083926843428237489276151990912312851155076104204473921520006924263781830121865419779825183
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/7
nats:
0.2879689768406531743495864745289245949304685580985984238016146144860968103549950215729736471889655101
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/7
bits:
0.4154514148178790878606219169569526095667530977930157305681782669150140552834574184632031426134576089
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/8
nats:
0.2578768678804410284670812798204655814498204034138262132709857771064678637814093553856688026199965247
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/8
bits:
0.3720376784510906839322589404121531684870023015453621741669567105993895492816098750585180642194144168
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/9
nats:
0.2340367889029837038973642912217222466333856327157239102993074056103664916334860464865778238696676316
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/9
bits:
0.3376437147359117721243279344893879434611972247448705470381887906504604333474689485946619167645672278
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/11
nats:
0.1984769516204080553121126384601660966861591045336269164834061742155717004752853935948755772267244065
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/11
bits:
0.2863417138335214113697194245064983127851207331293872038354911252831917721989848180203852447455796220
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/12
nats:
0.1848068533548667838970980442148743435876292032790375255488866387704763967710986288773761260445457313
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/12
bits:
0.2666199308573601990509035227675108114710121577276099693096457778709545363533619303922811868942613870
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/13
nats:
0.1730678885504889148860101405560906544038983244848068411260605549232483398722403974718682566220639473
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/13
bits:
0.2496841845489141669914593170409811633978098142285181284856797091856220723672646940096772097566206148
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/14
nats:
0.1628652385002697114375526350325987517112937451983234324989771294170053875833807881554510906540837545
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/14
bits:
0.2349648719175373887108750084124007733008707040599728382759547569870821690650685958582274153290504278
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/15
nats:
0.1539065163673436534496277476259856714609280371814112789829381691980798520515846411712705502912528582
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/15
bits:
0.2220401679236627680120666728644183443041379829509929857598590299760807164036183349160736863187645633
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/16
nats:
0.1459702625320655727114896791219884595574148633353337304458964400379146328214598225658524723107244915
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/16
bits:
0.2105905738722710646593532695231125480645221328642876838503929496343041632101897295598269621468942497
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/17
nats:
0.1388854656604912838321874174648908098733769902268352618909948885766482355881177797823355160592036577
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/17
bits:
0.2003693725599451959342491234199242584621985298526932587028946698977897012284125253779441850049141580
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/18
nats:
0.1325178361098633946219038708598028612331597782889659462683721089333159552943894952986353928869117370
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/18
bits:
0.1911828249850363216227716193228148703651737400903399229251565897900684060061014989833253164004412668
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/19
nats:
0.1267603580362998815193270153525176892970472175187954063148612469440882536279829924856300741034223678
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
1/19
bits:
0.1828765399202792988186339029660992710698636340529644170849859534582861665313157028941312368919522381
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
2/3
nats:
1.235683292859068699592046597689359648609048528700148626337109002836376385771132014770017820449131651
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
2/3
bits:
1.782714158717123062358425046973387841672760651346103682216678823943079724950977292072525967876155186
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
3/4
nats:
1.465373048643782336864505853990687129513557240995943415925093142917989316870698464070755688793138879
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
3/4
bits:
2.114086430330726522609809150230704876694781413892809147478674826223656876532774727912519455720457841
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
4/5
nats:
1.635881143719581678806271086756414292285703897491368574994849124739764244824171182117860592225377518
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
4/5
bits:
2.360077613528006114445490398375571927049637695455695079015828053905229893661589487881731713989754043
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
5/6
nats:
1.771407931273635740121731244279760351209859331205982605382104809412221235183966249267405803249093495
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
5/6
bits:
2.555601437839851995449757173738904447450540428789561230021239215562939889963667487694058805454692555
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
6/7
nats:
1.883805599344425438320616620472063133020380801685053757291181808454415519319739185216241586690439716
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
6/7
bits:
2.717756996173064075911905061617192203945912705884475589874784535009579834252609386506807258384957921
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
7/8
nats:
1.979773765474876494610222871820812217628985461340413607003760263356607120382483724206787975058832687
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
7/8
bits:
2.856209793532673995745089893017822156182820897841585353867693156613048676585818070604604217649368231
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
8/9
nats:
2.063471194303182936355487455042145926410692407430828903083355655598757913138670997386050388296463652
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
8/9
bits:
2.976959659038428662369508133584531716896875313567212399731250808578242019332077635388211092511425761
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
9/10
nats:
2.137656983403780625275665773469319524666743019098531520723532045420129878734397828407588975681090242
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
9/10
bits:
3.083987129078295461066664433250669807118398224403075028239804758483471989377569408096057352505076517
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
10/11
nats:
2.204255293829840840587157753306148599147910097238275838671804509589408064903449280971378734790609836
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
10/11
bits:
3.180068181261556281398650453786865309087580020045162814671104933993232177736457988691278555908260484
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
11/12
nats:
2.264661250976014497225604321946006725914857355418685638590271270374375445312197088020106447255400686
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
11/12
bits:
3.267215556076492256299100517344020323831998762091572867306664390868056748955182279057694992752984856
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
12/13
nats:
2.319918488098000600062957397504463949844896905439104339446642180405685351824811371433476156355344260
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
12/13
bits:
3.346934898045607143604735810210074308195029750037651242956421472615994420761382012566488714865897855
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
13/14
nats:
2.370827980081538331184643734650329685179122402324236334475878394702415558859327620818470853715671635
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
13/14
bits:
3.420381769664433465407494797279309981802799387676635816604459797536056611259650337127741367671422021
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
14/15
nats:
2.418017789338432341141707687005000557063630354919723363464064775384321129592735156777124246595700641
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
14/15
bits:
3.488462273459850553007934988676841177494006673966689974312427709935142542443229748882289504458771422
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
15/16
nats:
2.461989441723036386600936439269602156645276140091567700648921545580814916350641769407696671685972862
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
15/16
bits:
3.551899958294812171686962601349444244774423109029705618312548167491543928873380842119309301333358310
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
16/17
nats:
2.503149738505602775623864050254774728624829332275155287797623531097802749630660275042299938686710921
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
16/17
bits:
3.611281714344538657276826674075719772957550387862605662705148258275407596093118697021146491833657067
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
17/18
nats:
2.541833167667458054447462256854232484636337371651295392025568323940682842881802314474485715564780528
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
17/18
bits:
3.667090105760926778229537985694753610945484864016111163946368212577954342377784127953670502486712402
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
18/19
nats:
2.578318061334865694389100888068851368757273790786229800489730257514870176604378835812215830679667948
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
18/19
bits:
3.719726680922256925423189162595451458660231190409843928435404296284259901129234885386753149508617882
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
19/20
nats:
2.612838482236904137672911813330705115333050756571448392266388755574604303070048682462874810889067721
logarithmic series with parameter $p$ on $\{1,2,\ldots\}$
19/20
bits:
3.769529120967027504426639356036308624444156034752805893659501852946125344152611863060753438917308649
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
1, 10
nats:
1.993805759248670333856896957152200002515881919313234443059920928844971188262118361333737047027525301
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
1, 10
bits:
2.876453681363911178606028951957139062281746896843547256136816505700920738876608669920150793229488495
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
1, 100
nats:
3.680777745058364587712049705079120647154594445970691381230316568487131515270354840259475373983094601
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
1, 100
bits:
5.310239799410163827185483011220759943936550257290205835622511376440494925019213330118328210859206088
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
2, 10
nats:
1.236292605723536032606356156161327182303383428318270068948546189234125242762983312473959499140141832
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
2, 10
bits:
1.783593211365039932780342812340463340400287991633168443499177928725208975717239302939316395800895220
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
2, 100
nats:
1.570207647482339540162613521902729790654660931501783911692986346654237522483118263510973378003362037
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
2, 100
bits:
2.265330786188696882756995338066582442676759867526216877642268659046588803984839013505637568288465058
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
3, 10
nats:
0.6442558055995113636683707044605262008712346456536760077386418469395852710453585301768664047657016517
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
3, 10
bits:
0.9294646558023391069451494849023790272095154616243967468365773549318759638704603085592367034572303273
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
3, 100
nats:
0.6778500429477991599458253306331032238443757698120348972410214714176782424088989142075970344581527653
Zipfian with exponent $s$ on $\{1,\ldots,N\}$
3, 100
bits:
0.9779308954271607108279556980630223999262798214704322874823811143700003679754981901249003433872625061
Benford leading-digit distribution on $\{1,\ldots,9\}$
-
nats:
1.993433150791204269825208861305402097795519730691358225187074927657154761171872828137507030847518217
Benford leading-digit distribution on $\{1,\ldots,9\}$
-
bits:
2.875916120990131601775229975403237439422636996748531479535509615582896465138313864332778023188451518
Definition
For a discrete random variable $X$ with probability mass function $p$, the Shannon entropy [1] is $H(X)=-\sum_x p(x)\log p(x)$. This table gives $H(X)$ for named discrete distributions in SciPy standard forms, in the two logarithmic units nats and bits.
Parameters
distribution
—   distribution family, in standard form
shape
—   shape parameter or tuple of shape parameters (the value is - for a distribution with no shape parameter; otherwise it is the distribution's parameters, in the order its label names them, separated by commas)
unit
—   logarithmic unit
Formulas
(1)
$H_{\mathrm{bits}}(X)=H_{\mathrm{nats}}(X)/\log 2$.
(2)
$H(\operatorname{Bernoulli}(p))=H(\operatorname{Bernoulli}(1-p))$.
(3)
$H(\operatorname{Binom}(n,p))=H(\operatorname{Binom}(n,1-p))$.
(4)
The uniform distribution on $n$ points has entropy $\log n$ nats.
(5)
For the geometric distribution on positive integers, $H=-\log p-\frac{1-p}{p}\log(1-p)$ nats.
(6)
For the hypergeometric distribution, write $H(N,K,n)$ for the entropy with population $N$, successes $K$, and draws $n$; then $H(N,K,n)=H(N,n,K)=H(N,K,N-n)$.
(7)
Benford's law has probabilities $p_d=\log_{10}(1+1/d)$ for $1\leq d\leq9$.
(8)
For integers $1\leq a<b$, $\log b=H(\operatorname{Bernoulli}(a/b))+(a/b)\log a+((b-a)/b)\log(b-a)$ in nats.
(9)
For success probability $0<q<1$, $H(\operatorname{Geom}(q))=H(\operatorname{Bernoulli}(q))/q$.
(10)
For $r>0$ and $0<q<1$, let $N\sim\operatorname{Poisson}(-r\log q)$ be independent of a sequence of independent, identically distributed jumps $U_i$ with $\Pr(U_i=k)=-(1-q)^k/(k\log q)$ for $k\geq1$. Then $\sum_{i=1}^{N}U_i\sim\operatorname{NB}(r,q)$ in the failures-before-success convention. The jump parameter is $p=1-q$, and its mean is $-p/((1-p)\log(1-p))$.
Comments
(11)
The support and parameter order are those used by SciPy [2]. In particular, the geometric and logarithmic series distributions are supported on positive integers, and the negative binomial row counts failures before the $r$th success.
(12)
Rows whose bit value is an exact rational number are omitted; each is named in the comment on the same distribution's nats row. These are the Bernoulli$(1/2)$ row in bits, the geometric$(1/2)$ row in bits, and the bits rows for $\operatorname{Binom}(2,1/2)$ and for uniform distributions on $2^m$ points.
(13)
The Bernoulli rows are the binary entropy function $H_2(p)$.
(14)
A binary partition of a uniform $b$-outcome model into parts of sizes $a$ and $b-a$ has Bernoulli probability $a/b$. The entropy of the partition is unchanged on exchanging its parts [3]. The finite selection is exploratory, not a claim of documented use for every partition.
(15)
Independent repetitions of a Bernoulli experiment give the positive-support geometric waiting time [4]. One marked outcome among $m$ equally likely outcomes gives success $q=1/m$; declaring all other outcomes successes gives $q=1-1/m$. Only the first orientation has integer mean waiting time $m$.
(16)
The logarithmic-series parameter $p$ controls the decay of its probability mass function; it is not a Bernoulli success probability. In the compound-Poisson representation of a negative-binomial distribution, it equals one minus the negative-binomial success probability. This representation is used in Bayesian count-regression inference [5]. The model parameter $r$ controls the Poisson rate, not the jump law.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

R = RealBallField(numberdb.bits(100, losing=96))
p = R(QQ(1) / QQ(3))
# Entropy in nats of Bernoulli(1/3).
-p * p.log() - (1 - p) * (1 - p).log()
Links
Similar tables
Differential entropies of continuous probability distributions —   the same entropy functional for absolutely continuous distributions
Entropy constants of lattice models —   entropy per site in lattice models is a limiting entropy rather than the entropy of one named distribution
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds Bernoulli$(p)$ for reduced $0<p\leq1/2$ with denominator at most $12$, together with $p=1/m$ for $2\leq m\leq20$; binomial $\operatorname{Binom}(n,p)$ for $2\leq n\leq20$ and $p\in\{1/2,1/3,1/4,1/10\}$; geometric success probabilities and logarithmic-series decay parameters in $\{1/m,1-1/m:2\leq m\leq20\}$; negative binomial rows for $2\leq r\leq10$ and $p\in\{1/2,1/3,1/4,1/10\}$; Poisson$(\lambda)$ for $\lambda=1/2,1,3/2,\ldots,10,11,\ldots,20$; uniform rows for $2\leq n\leq30$; hypergeometric rows with $(N,K,n)$ in $\{10\}\times\{2,5\}\times\{2,5,8\}$ and $\{20\}\times\{2,5,10\}\times\{2,5,10\}$; Zipfian rows with $s\in\{1,2,3\}$ and $N\in\{10,100\}$; and Benford's leading-digit law, each in nats and in bits except where the bit value is an exact rational number. These are 589 selected entries, not an assertion of complete coverage.

Bernoulli symmetry and its entropy maximum at $p=1/2$ are intrinsic [3]. The denominator-12 selection is a bounded exploratory reference family of binary partitions of small uniform models. The additional reciprocal probabilities connect the Bernoulli indicators to geometric waiting times. Geometric success $q=1/m$ gives integer mean waiting time $m$; $q=1-1/m$ gives the frequent-success companion with mean $m/(m-1)$ [4]. Both orientations are retained because geometric entropy has no complement symmetry.

For logarithmic-series entropy, $p=1-q$ is the jump parameter in the compound-Poisson representation of a negative-binomial law with success probability $q$ [5]. This relationship, rather than a probability mesh, selects the two rational orientations and includes inputs above $1/2$. The references establish the distributions and relationships, not demand for each selected input. The bounds $12$ and $20$ are noncanonical reference choices that balance small-model breadth and the cost of the logarithmic tail at $p=19/20$.

Integer trial, success, support and urn counts are discrete families with noncanonical finite cutoffs; the selected Poisson rates and truncated Zipf slices are exploratory reference choices. Base-ten Benford probabilities are fixed by the stated leading-digit convention.)
How they were obtained:

Offline verification completed at 2026-11-01T22:56:47.314613+00:00 recomputed all 589 selected entries in SageMath 10.9 real ball arithmetic at 100 and 150 requested significant decimal digits, using working precision numberdb.bits(digits, losing=128). Every enclosure was finite, positive, sufficiently narrow and wholly contained in the interval denoted by the stored 100-digit decimal, plus or minus one unit in its last place. The 451 retained numeric strings and row comments were preserved unchanged; the other 138 values were newly computed.

more

Bernoulli, binomial, hypergeometric and truncated Zipf probabilities are constructed exactly before ball evaluation. Benford probabilities are enclosed by logarithm ratios in ball arithmetic, not treated as exact rationals. Geometric and uniform entropies use closed forms. Negative-binomial, Poisson and logarithmic-series entropies use positive series-tail bounds derived in the repaired source. The tail bound is below $10^{-d-12}$ nats for $d$ requested digits and is added before conversion to bits by division by $\log 2$. These enclosures, not agreement with floating-point software, justify the recorded precision.

At 100 requested digits the largest enclosure radius was 1.237233e-12 stored last-place units, with at least 111 relative decimal digits across all entries. At 150 requested digits the corresponding figures were 1.398842e-62 and 161. For the new logarithmic-series endpoint $p=19/20$, the direct entropy sums used 5065 and 7310 terms respectively, with certified tails below $10^{-112}$ and $10^{-162}$ nats before conversion to bits.

Independent expectation-identity computations at 170 requested digits checked all 206 logarithmic-series, negative-binomial and Poisson rows against their stored intervals and the direct 150-digit calculation. Known-answer controls, probability-complement identities, the geometric and Bernoulli entropy relationship, and nats/bits conversions also passed. The offline verify(sample=None) matched all 589 rows separately at 100 and 150 requested digits.

An additional comparison of all 589 rows with SciPy 1.17.1 found maximum absolute differences 4.329386e-10 nats and 6.245982e-10 bits. These are double-precision convention checks, not certificates of the stored precision. This fresh verification neither identifies the historical producer of retained values nor attributes the new values to the earlier 717-row verification.



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