History of Shannon entropies of discrete probability distributions

 Definition: For a discrete random variable $X$ with probability mass function $p$,   the Shannon entropy CITE{WikiShannonEntropy} is $H(X)=-\sum_x p(x)\log p(x)$. This-  table gives $H(X)$ for named discrete distributions in SciPy standard forms; nats-  use natural logarithms, and bits divide by $\log 2$.+  table gives $H(X)$ for named discrete distributions in SciPy standard forms, in+  the two logarithmic units nats and bits. Tags: - probability theory

       binomial: binomial $\operatorname{Binom}(n,p)$       geometric: geometric with success probability $p$ on $\{1,2,\ldots\}$-      negative-binomial: negative binomial counting failures before $r$ successes+      negative-binomial: negative binomial $\operatorname{NB}(r,p)$, failures before+        the $r$th success with success probability $p$       poisson: Poisson$(\lambda)$       discrete-uniform: uniform on $\{1,\ldots,n\}$

         $n$       logarithmic: logarithmic series with parameter $p$ on $\{1,2,\ldots\}$-      zipf: Zipfian on $\{1,\ldots,N\}$ with exponent $s$+      zipf: Zipfian with exponent $s$ on $\{1,\ldots,N\}$       benford: Benford leading-digit distribution on $\{1,\ldots,9\}$   shape:

     display: shape     constraints: the value is `-` for a distribution with no shape parameter; otherwise-      it is the displayed integer, rational, half-integer or comma-separated tuple-      named by the distribution+      it is the distribution's parameters, in the order its label names them, separated+      by commas   unit:     type: Symbolic

     on positive integers, and the negative binomial row counts failures before the     $r$th success.-  omitted-rational-bits: 'Rows whose bit value is an exact rational number are omitted:-    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.'+  omitted-rational-bits: 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.   bernoulli: The Bernoulli rows are the binary entropy function $H_2(p)$. Formulas:

   formula-geometric: For the geometric distribution on positive integers, $H=-\log     p-\frac{1-p}{p}\log(1-p)$ nats.+  formula-hypergeometric-symmetry: For the hypergeometric distribution, write $H(N,K,n)$+    for the entropy with population $N$, successes $K$, and draws $n$; then $H(N,K,n)=H(N,n,K)=H(N,K,N-n)$.   formula-benford: Benford's law has probabilities $p_d=\log_{10}(1+1/d)$ for $1\leq     d\leq9$.

     title: 'SciPy: Statistical functions'     url: https://docs.scipy.org/doc/scipy/reference/stats.html-  RequestedIssue55:-    title: Requested in numberdb-data#55-    url: https://github.com/numberdb/numberdb-data/issues/55 Programs:   program-sage:

   type: R   rigour: proven-  complete: 'false'+  complete: 'no'   complete-note: it holds Bernoulli$(p)$ for reduced $p\leq1/2$ with denominator at-    most $12$, together with $p=0.01,\ldots,0.11$; binomial $\operatorname{Binom}(n,p)$-    for $2\leq n\leq20$ and $p\in\{1/2,1/3,1/4,1/10\}$; geometric and logarithmic-    series rows for $p\in\{1/2,1/3,1/4,1/5,1/10,11/100\}$; negative binomial rows-    for $2\leq r\leq10$ and $p\in\{1/2,1/3,1/4,1/10\}$; Poisson$(\lambda)$ for $\lambda=1/2,1,3/2,\ldots,10,11,\ldots,20$;+    most $12$, together with every two-decimal $p$ with $0.01\leq p\leq0.50$; binomial+    $\operatorname{Binom}(n,p)$ for $2\leq n\leq20$ and $p\in\{1/2,1/3,1/4,1/10\}$;+    geometric and logarithmic series rows for $p\in\{1/2,1/3,1/4,1/5\}$ and every+    two-decimal $p$ with $0.01\leq p\leq0.50$; 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 $s\in\{1,2,3\}$

     uniform rows used the closed forms in Formulas. Infinite logarithmic series, negative     binomial and Poisson rows were summed with explicit positive tail bounds and the-    bound was added to the returned ball. All 469 entries were compared with SciPy+    bound was added to the returned ball. All 717 entries were compared with SciPy     1.17.1 `stats` entropy methods before the draft was filled; the largest difference     was $3.32\cdot10^{-10}$ bits, on the negative binomial row with $r=9$ and $p=1/10$.

   number-header: $H(X)$ Numbers:-- comment: The omitted bits row is exactly $1$.-  params:-    distribution: bernoulli-    shape: 1/2-    unit: nats-  number: '0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875'-- params:-    distribution: bernoulli-    shape: 1/3-    unit: nats-  number: '0.6365141682948128184504238226170746592638238015825792823209076606418985452388124898030402034827862971'-- params:-    distribution: bernoulli-    shape: 1/3-    unit: bits-  number: '0.9182958340544895147870722772811498420931477410258143937890859878744315611276918958556138082514215754'-- params:-    distribution: bernoulli-    shape: 1/4-    unit: nats-  number: '0.5623351446188083502880303152244588576653823503534484194403392687586665240254327060565148378825383085'-- params:-    distribution: bernoulli-    shape: 1/4-    unit: bits-  number: '0.8112781244591328639096957920391376184301391942306392046581855090941763291542310781082896438114338184'-- params:-    distribution: bernoulli-    shape: 1/5-    unit: nats-  number: '0.5004024235381878795331879388931051306048011392921093153195598762847491925561462196607525548989097107'-- params:-    distribution: bernoulli-    shape: 1/5-    unit: bits-  number: '0.7219280948873623478703194294893901758648313930245806120547563958159347766086252158501397433593701551'-- params:-    distribution: bernoulli-    shape: 1/10-    unit: nats-  number: '0.3250829733914482395065500282238179392356184845478239629108781004200828818838563398634888464248805385'-- params:-    distribution: bernoulli-    shape: 1/10-    unit: bits-  number: '0.4689955935892812212535893303833204600971654591781147032344016176419579665787798033100348885068113193'-- params:-    distribution: bernoulli-    shape: 11/100-    unit: nats-  number: '0.3465153369186661520863132845963983869516674787760962287376525101445031561505328626958518310638733583'-- params:-    distribution: bernoulli-    shape: 11/100-    unit: bits-  number: '0.4999159581645279956404995941302756626364007555431767231334216282975329506755548763322804854150763222'-- params:-    distribution: bernoulli-    shape: 2/5-    unit: nats-  number: '0.6730116670092564359967193424894015895069069658307659492235592874943249629886144982636190944063728825'-- params:-    distribution: bernoulli-    shape: 2/5-    unit: bits-  number: '0.9709505944546686389980760631207002706089427484090919757813048030912758399320100783367714584085172098'-- params:-    distribution: bernoulli-    shape: 1/6-    unit: nats-  number: '0.4505612088663046886451779140255459064516562302925732709281677669024054254319222119543675167325011007'-- params:-    distribution: bernoulli-    shape: 1/6-    unit: bits-  number: '0.6500224216483542248951394193733246955391215801719972170767889913611525806205042159804973554519464462'-- params:-    distribution: bernoulli-    shape: 1/7-    unit: nats-  number: '0.4101163182884090186946578648311492101602355648535714405833549986825330783049518256623907141791708616'-- params:-    distribution: bernoulli-    shape: 1/7-    unit: bits-  number: '0.5916727785823273804816216509908452297040428479440141604295037344636652250384570757644777349547065776'-- params:-    distribution: bernoulli-    shape: 2/7-    unit: nats-  number: '0.5982695885852572348427440421507095390972265810014184573444473676007681223981152314501733493794807514'-- params:-    distribution: bernoulli-    shape: 2/7-    unit: bits-  number: '0.8631205685666310018203125818822663973090042023771546322096085842017960084274606894620468967706606743'-- params:-    distribution: bernoulli-    shape: 3/7-    unit: nats-  number: '0.6829081047004716551734109316670383498447314798175339901494582818147753119787286070065508013860792746'-- params:-    distribution: bernoulli-    shape: 3/7-    unit: bits-  number: '0.9852281360342514582475097698256237334582490226693631863391120149812787512360393168454550813481729671'-- params:-    distribution: bernoulli-    shape: 1/8-    unit: nats-  number: '0.3767701612564367862845127138617474407940512646966372224600736472847984860010235375033887947248684765'-- params:-    distribution: bernoulli-    shape: 1/8-    unit: bits-  number: '0.5435644431995964059882768474221480424391017022796268142823697414384692572455811368143716257260655630'-- params:-    distribution: bernoulli-    shape: 3/8-    unit: nats-  number: '0.6615632381579820598530048172622152902801905974794111417661247211947586386348196813461503991258692726'-- params:-    distribution: bernoulli-    shape: 3/8-    unit: bits-  number: '0.9544340029249649645358982525886999492995499764749567197948700071621289291967247791478074823061105623'-- params:-    distribution: bernoulli-    shape: 1/9-    unit: nats-  number: '0.3488320958430318910112048166232472277603140906848182258142419752926055945180320254649293043036810108'-- params:-    distribution: bernoulli-    shape: 1/9-    unit: bits-  number: '0.5032583347756456962408112212289663508529621487182954542448386424155297889220504583778942831695098175'-- params:-    distribution: bernoulli-    shape: 2/9-    unit: nats-  number: '0.5297061990576545211713867575096501600049151850017723670852674374361167771923430108888608269488043927'-- params:-    distribution: bernoulli-    shape: 2/9-    unit: bits-  number: '0.7642045065086202793415017522708757219099414396335192891625006348052802398070114688795579504815680958'-- params:-    distribution: bernoulli-    shape: 4/9-    unit: nats-  number: '0.6869615765973233440858622918676791046025357993983177209662020524618725959599016506363897311344366715'-- params:-    distribution: bernoulli-    shape: 4/9-    unit: bits-  number: '0.9910760598382221696461893159570829198169447081490840031033073114066771352505920051278166479698597313'-- params:-    distribution: bernoulli-    shape: 3/10-    unit: nats-  number: '0.6108643020548934630256709631973806854608950105746453085913464959200184177853213027225515296339686028'-- params:-    distribution: bernoulli-    shape: 3/10-    unit: bits-  number: '0.8812908992306926182248192242427636571881684325405377453439263926043807140667825565449529014647961329'-- params:-    distribution: bernoulli-    shape: 1/11-    unit: nats-  number: '0.3046360973492381040455877100702527474570691370021690151882696173418712514792799128605117942848181566'-- params:-    distribution: bernoulli-    shape: 1/11-    unit: bits-  number: '0.4394969869215133035899817471899837079170211683866947876266594677118444591667227019366369871420185296'-- params:-    distribution: bernoulli-    shape: 2/11-    unit: nats-  number: '0.4741393130578373564302273500086005889302677348892352989034897069067841219246593644444355952171788848'-- params:-    distribution: bernoulli-    shape: 2/11-    unit: bits-  number: '0.6840384356390416865477902293566386716417170403667990632942973019318062469656134322184868857845742744'-- params:-    distribution: bernoulli-    shape: 3/11-    unit: nats-  number: '0.5859526183035508622256429756229643227531182268361104066639855974256709141924639789694576977885216307'-- params:-    distribution: bernoulli-    shape: 3/11-    unit: bits-  number: '0.8453509366224364794392524256491157290414639599474550548248691035787583757761023811125057152183309682'-- params:-    distribution: bernoulli-    shape: 4/11-    unit: nats-  number: '0.6554817739013927612369139256226137859977437464869562341111885159023728952748692040422805165402302288'-- params:-    distribution: bernoulli-    shape: 4/11-    unit: bits-  number: '0.9456603046006400969181098448509915350225782182487693907908887304088537158986228158472161179967663528'-- params:-    distribution: bernoulli-    shape: 5/11-    unit: nats-  number: '0.6890092384766585551638835037455700421884385880791792802462508440254624939434084477998234460638345449'-- params:-    distribution: bernoulli-    shape: 5/11-    unit: bits-  number: '0.9940302114769564536471784275318066012593184882928781237665927451512157788282658568563657023500477407'-- params:-    distribution: bernoulli-    shape: 1/12-    unit: nats-  number: '0.2868359830561606448395945333708436492952595437672942160257006192545890485708647444332089264708209051'-- params:-    distribution: bernoulli-    shape: 1/12-    unit: bits-  number: '0.4138168503036336966043228177825062966151855091508603284190178126252951597176750725385801172384294608'-- params:-    distribution: bernoulli-    shape: 5/12-    unit: nats-  number: '0.6791932659915257261679373239861124820411908366752918825778068103797853723411005899466525020914862353'-- params:-    distribution: bernoulli-    shape: 5/12-    unit: bits-  number: '0.9798687566511528071666237466086692971088691287853236816211839839101049090378188137943033323357277195'-- params:-    distribution: bernoulli-    shape: 1/20-    unit: nats-  number: '0.1985152433458725564266475158490799153010913247548558525429456858721381646860597715634100674610640849'-- params:-    distribution: bernoulli-    shape: 1/20-    unit: bits-  number: '0.2863969571159561287664759777278974743059992018461232982023079014945978288134335733875670918958024613'-- params:-    distribution: bernoulli-    shape: 1/25-    unit: nats-  number: '0.1679441477341729543404547292072020865321713860696608382924307953153898226527301401167102498154905962'-- params:-    distribution: bernoulli-    shape: 1/25-    unit: bits-  number: '0.2422921890824147615450494727888765033202409546643794060719902432724152545346662116788902307973755978'-- params:-    distribution: bernoulli-    shape: 2/25-    unit: nats-  number: '0.2787693717685874089059274914703407246381728081350141408568968349045326993517255176692696852458904974'-- params:-    distribution: bernoulli-    shape: 2/25-    unit: bits-  number: '0.4021791902022728532300224743491250147907436706420105693746245581782736592995069112783853229256191531'-- params:-    distribution: bernoulli-    shape: 1/50-    unit: nats-  number: '0.09803911327973198061225941076191957703801677291684276856557109856409506639095448000232042595054971624'-- params:-    distribution: bernoulli-    shape: 1/50-    unit: bits-  number: '0.1414405425418206451543789972043919667932505991555252881020210124540406894473521781644719283451271714'-- params:-    distribution: bernoulli-    shape: 3/50-    unit: nats-  number: '0.2269675225006044055233424441093570184855659375289151396334162864250928190964406985996657696652927617'-- params:-    distribution: bernoulli-    shape: 3/50-    unit: bits-  number: '0.3274449191544761950128524133651065220106155539650904498938617310481118099811849762154853731295092786'-- params:-    distribution: bernoulli-    shape: 1/100-    unit: nats-  number: '0.05600153435484734045207319807664951317668188737045903416557898929363863110775486011793105472539861738'-- params:-    distribution: bernoulli-    shape: 1/100-    unit: bits-  number: '0.08079313589591117282486633370356863526903104839309833380744890637142774866160230872376776008689410701'-- params:-    distribution: bernoulli-    shape: 3/100-    unit: nats-  number: '0.1347421681797667398409055792082097816681308049438455532176676342286484175333051798837716748515383206'-- params:-    distribution: bernoulli-    shape: 3/100-    unit: bits-  number: '0.1943918578315761608655943296458712861413420598012773323289368456381923157762491710228600763858658681'-- params:-    distribution: bernoulli-    shape: 7/100-    unit: nats-  number: '0.2536389469216914149669073251653647230869976220647081450274849537549122188330311265755013701587141808'-- params:-    distribution: bernoulli-    shape: 7/100-    unit: bits-  number: '0.3659236509002232152780288416678172280775727639928899609816645906900039985197769117370448800692025383'-- params:-    distribution: bernoulli-    shape: 9/100-    unit: nats-  number: '0.3025378230974980861302941983649403040231137431981688933150343707334457242057590494943613968669162964'-- params:-    distribution: bernoulli-    shape: 9/100-    unit: bits-  number: '0.4364698170641029793456742550136151366876848374054825138011400363934816799136172495991028494327416565'-- comment: The omitted bits row is exactly $3/2$.-  params:-    distribution: binomial-    shape: 2,1/2-    unit: nats-  number: '1.039720770839917964125848182187264852113250201540382881181020014240090432954542073408794990494628031'-- params:-    distribution: binomial-    shape: 2,1/3-    unit: nats-  number: '0.9649629230074277216043000356971819549385364323383784516992908726200665918244273282256967060782753997'-- params:-    distribution: binomial-    shape: 2,1/3-    unit: bits-  number: '1.392147223664534585129700110117855239741851037607184343133727531304418677810939347266783172058398706'-- params:-    distribution: binomial-    shape: 2,1/4-    unit: nats-  number: '0.8647400965276372095445985849021015023024521503218011185854235339573104398122298937608309281414196092'-- params:-    distribution: binomial-    shape: 2,1/4-    unit: bits-  number: '1.247556248918265727819391584078275236860278388461278409316371018188352658308462156216579287622867637'-- params:-    distribution: binomial-    shape: 2,1/10-    unit: nats-  number: '0.5253994542821063233179982745851640962176469449108019800800337991313549118131676309179222939904057133'-- params:-    distribution: binomial-    shape: 2,1/10-    unit: bits-  number: '0.7579911871785624425071786607666409201943309183562294064688032352839159331575596066200697770136226387'-- params:-    distribution: binomial-    shape: 3,1/2-    unit: nats-  number: '1.255482325178753659705262436682635425740882484713703673561019278252060145995127421662378164878956996'-- params:-    distribution: binomial-    shape: 3,1/2-    unit: bits-  number: '1.811278124459132863909695792039137618430139194230639204658185509094176329154231078108289643811433818'-- params:-    distribution: binomial-    shape: 3,1/3-    unit: nats-  number: '1.177134312439031994421107976569540174693144366199238212472926759500699440237364824826710107239204165'-- params:-    distribution: binomial-    shape: 3,1/3-    unit: bits-  number: '1.698245835016031090058724202544905187106233617949122474396756193929229198186836645885321108142205898'-- params:-    distribution: binomial-    shape: 3,1/4-    unit: nats-  number: '1.069036021480613349454265499904455864131933612285048691720252243604909032140830574303135651564890626'-- params:-    distribution: binomial-    shape: 3,1/4-    unit: bits-  number: '1.542292966721748239661359220146766069113021978364897017468195659103161234328366542906086164292876819'-- params:-    distribution: binomial-    shape: 3,1/10-    unit: nats-  number: '0.6786236022339551018429338707023718774520330030313295367642668311781251864825445985345902854749339516'-- params:-    distribution: binomial-    shape: 3,1/10-    unit: bits-  number: '0.9790469055731314947682584762840509229263464874573742233801516361997773782318625980490889372925501326'-- params:-    distribution: binomial-    shape: 4,1/2-    unit: nats-  number: 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'0.6199210138989939431029699701850065660660693465803148960494820689385892689550772480352166469630503694'-- params:-    distribution: logarithmic-    shape: 27/100-    unit: nats-  number: '0.4936850390339537317330952820441637604398916479917606364624670194117657194931778699856117526455603508'-- params:-    distribution: logarithmic-    shape: 27/100-    unit: bits-  number: '0.7122369575753593748230680954840349797181108488158631574294459024408022994772175105026679497611179059'-- params:-    distribution: logarithmic-    shape: 29/100-    unit: nats-  number: '0.5257414053633178019925175436371555239826635736846395588619543075476162831504478005947826727545736626'-- params:-    distribution: logarithmic-    shape: 29/100-    unit: bits-  number: '0.7584845183076528619802775510162062422309501469973828132892071559479788780787712024343815932946783298'-- params:-    distribution: logarithmic-    shape: 31/100-    unit: nats-  number: '0.5579319785949505135369295931131409172026152634132130141328299616069250735222189941881185398935726253'-- params:-    distribution: logarithmic-    shape: 31/100-    unit: bits-  number: '0.8049256986723023876920129517402771872560329025427229130687603148897466010321169268647441703460884054'-- params:-    distribution: logarithmic-    shape: 33/100-    unit: nats-  number: '0.5903221777248257466461536571808776910741557630477783700887052778662747073528179015360567227823248422'-- params:-    distribution: logarithmic-    shape: 33/100-    unit: bits-  number: '0.8516548783303794041010946420185463896208158013686895297779323209930400357025604185052468346493329570'-- params:-    distribution: logarithmic-    shape: 37/100-    unit: nats-  number: '0.6559560185185333927951651295174023011584029414519068908824843072709509496208038732664915621983223444'-- params:-    distribution: logarithmic-    shape: 37/100-    unit: bits-  number: '0.9463444949579571711604417665547275231517204845868417583804417219114234957940863094512144657469063547'-- params:-    distribution: logarithmic-    shape: 39/100-    unit: nats-  number: '0.6893267444971231309445123010826161686908105342462054168165371289180836128990393514380441919331292942'-- params:-    distribution: logarithmic-    shape: 39/100-    unit: bits-  number: '0.9944882758381330868554052007930176091732237474959195359993923902851930272669911343055800430877503467'-- params:-    distribution: logarithmic-    shape: 41/100-    unit: nats-  number: '0.7231532740853972323549196518581580960307685923776401690115743321919365628635903644046999636694023922'-- params:-    distribution: logarithmic-    shape: 41/100-    unit: bits-  number: '1.043289642325619922122610345447551946595241511762883512839218453299255752550231967296416157406819940'-- params:-    distribution: logarithmic-    shape: 43/100-    unit: nats-  number: '0.7575034052242885789781145110031467698519779035073407296568576156539687156488638533293177879005843673'-- params:-    distribution: logarithmic-    shape: 43/100-    unit: bits-  number: '1.092846406173584028615303419379037624179444199997591291656635439289035168112826319372095618507388848'-- params:-    distribution: logarithmic-    shape: 47/100-    unit: nats-  number: '0.8280639378227940956229204559385638814317483333873801344032833238391980953083034576450890780258844826'-- params:-    distribution: logarithmic-    shape: 47/100-    unit: bits-  number: '1.194643736635931980375300902271579622430044028226807386302256547392158283955813173929026003489046272'-- params:-    distribution: logarithmic-    shape: 49/100-    unit: nats-  number: '0.8644314323527175624375302741190666494470979329329788907385670492553900799881937764468616512863430619'-- params:-    distribution: logarithmic-    shape: 49/100-    unit: bits-  number: '1.247110940643809069382807273872180532942625756135771122949497122115675415452383162591964642229955681'+  bernoulli:+    1/2:+      nats:+        comment: In bits this entropy is exactly $1$.+        number: '0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875'+    1/3:+      nats: '0.6365141682948128184504238226170746592638238015825792823209076606418985452388124898030402034827862971'+      bits: '0.9182958340544895147870722772811498420931477410258143937890859878744315611276918958556138082514215754'+    1/4:+      nats: '0.5623351446188083502880303152244588576653823503534484194403392687586665240254327060565148378825383085'+      bits: '0.8112781244591328639096957920391376184301391942306392046581855090941763291542310781082896438114338184'+    1/5:+      nats: '0.5004024235381878795331879388931051306048011392921093153195598762847491925561462196607525548989097107'+      bits: '0.7219280948873623478703194294893901758648313930245806120547563958159347766086252158501397433593701551'+    1/10:+      nats: '0.3250829733914482395065500282238179392356184845478239629108781004200828818838563398634888464248805385'+      bits: '0.4689955935892812212535893303833204600971654591781147032344016176419579665787798033100348885068113193'+    11/100:+      nats: '0.3465153369186661520863132845963983869516674787760962287376525101445031561505328626958518310638733583'+      bits: '0.4999159581645279956404995941302756626364007555431767231334216282975329506755548763322804854150763222'+    2/5:+      nats: '0.6730116670092564359967193424894015895069069658307659492235592874943249629886144982636190944063728825'+      bits: '0.9709505944546686389980760631207002706089427484090919757813048030912758399320100783367714584085172098'+    1/6:+      nats: '0.4505612088663046886451779140255459064516562302925732709281677669024054254319222119543675167325011007'+      bits: 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'0.9910760598382221696461893159570829198169447081490840031033073114066771352505920051278166479698597313'+    3/10:+      nats: '0.6108643020548934630256709631973806854608950105746453085913464959200184177853213027225515296339686028'+      bits: '0.8812908992306926182248192242427636571881684325405377453439263926043807140667825565449529014647961329'+    1/11:+      nats: '0.3046360973492381040455877100702527474570691370021690151882696173418712514792799128605117942848181566'+      bits: '0.4394969869215133035899817471899837079170211683866947876266594677118444591667227019366369871420185296'+    2/11:+      nats: '0.4741393130578373564302273500086005889302677348892352989034897069067841219246593644444355952171788848'+      bits: '0.6840384356390416865477902293566386716417170403667990632942973019318062469656134322184868857845742744'+    3/11:+      nats: '0.5859526183035508622256429756229643227531182268361104066639855974256709141924639789694576977885216307'+      bits: 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'0.9798687566511528071666237466086692971088691287853236816211839839101049090378188137943033323357277195'+    1/20:+      nats: '0.1985152433458725564266475158490799153010913247548558525429456858721381646860597715634100674610640849'+      bits: '0.2863969571159561287664759777278974743059992018461232982023079014945978288134335733875670918958024613'+    1/25:+      nats: '0.1679441477341729543404547292072020865321713860696608382924307953153898226527301401167102498154905962'+      bits: '0.2422921890824147615450494727888765033202409546643794060719902432724152545346662116788902307973755978'+    2/25:+      nats: '0.2787693717685874089059274914703407246381728081350141408568968349045326993517255176692696852458904974'+      bits: '0.4021791902022728532300224743491250147907436706420105693746245581782736592995069112783853229256191531'+    1/50:+      nats: '0.09803911327973198061225941076191957703801677291684276856557109856409506639095448000232042595054971624'+      bits: 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'4.201887394001214751472641352896721724969469810372668779240524121494806876950183971882121389614160943'+  discrete-uniform:+    '2':+      nats:+        comment: In bits this entropy is exactly $1$.+        number: '0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875'+    '3':+      nats: '1.098612288668109691395245236922525704647490557822749451734694333637494293218608966873615754813732089'+      bits: '1.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242'+    '4':+      nats:+        comment: In bits this entropy is exactly $2$.+        number: '1.386294361119890618834464242916353136151000268720510508241360018986787243939389431211726653992837375'+    '5':+      nats: '1.609437912434100374600759333226187639525601354268517721912647891474178987707657764630133878093179611'+      bits: '2.321928094887362347870319429489390175864831393024580612054756395815934776608625215850139743359370155'+    '6':+      nats: '1.791759469228055000812477358380702272722990692183004705855374343130887915188303682479479081810150776'+      bits: '2.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242'+    '7':+      nats: '1.945910149055313305105352743443179729637084729581861188459390149937579862752069267787658498587871527'+      bits: '2.807354922057604107441969317231830808641026625966140783677291724070320848862192986497860999170210785'+    '8':+      nats:+        comment: In bits this entropy is exactly $3$.+        number: '2.079441541679835928251696364374529704226500403080765762362040028480180865909084146817589980989256063'+    '9':+      nats: '2.197224577336219382790490473845051409294981115645498903469388667274988586437217933747231509627464178'+      bits: '3.169925001442312362907477887895633017519628815384962120911505309082196455588717125044560949836176484'+    '10':+      nats: '2.302585092994045684017991454684364207601101488628772976033327900967572609677352480235997205089598298'+      bits: '3.321928094887362347870319429489390175864831393024580612054756395815934776608625215850139743359370155'+    '11':+      nats: '2.397895272798370544061943577965129299821706853937417175218567709130573623913236713075054708002634791'+      bits: '3.459431618637297256199363046725792958703231525681768071312801645726330619720018352709491299286900489'+    '12':+      nats: '2.484906649788000310229709479838878840798490826543259959976054352624281537157998398085342408806569464'+      bits: '3.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242'+    '13':+      nats: '2.564949357461536736053487441565318604805267944760207116419045510663464667324410179399574663440489489'+      bits: '3.700439718141092160396812654256694733628436401791037369538463525842855186633025300147376530281154896'+    '14':+      nats: '2.639057329615258614522584864901356297712584863942116442580070159430973484721763983393521825584290215'+      bits: '3.807354922057604107441969317231830808641026625966140783677291724070320848862192986497860999170210785'+    '15':+      nats: '2.708050201102210065996004570148713344173091912091267173647342225111673280926266731503749632906911700'+      bits: '3.906890595608518529324058373437206684624645800717061672510509050357033004402983778372420218277458397'+    '16':+      nats:+        comment: In bits this entropy is exactly $4$.+        number: '2.772588722239781237668928485832706272302000537441021016482720037973574487878778862423453307985674750'+    '17':+      nats: '2.833213344056216080249534617873126535588203012585744787297237737882292575800931280912094868037502948'+      bits: '4.087462841250339408254066010810404354011267282344820688126609064386696509047382068297343151843684273'+    '18':+      nats: '2.890371757896164692207722595303227977370481250005754157590068676768382208406912649353094836623882865'+      bits: '4.169925001442312362907477887895633017519628815384962120911505309082196455588717125044560949836176484'+    '19':+      nats: '2.944438979166440460009027431887853537237379261299128818537960236409292702064197288714158383815739896'+      bits: '4.247927513443585493793519422906834422693507569661534014581524730864565208205464886802708054172176520'+    '20':+      nats: '2.995732273553990993435223576142540775676601622989028230154007910460966231647047195841860532086016986'+      bits: '4.321928094887362347870319429489390175864831393024580612054756395815934776608625215850139743359370155'+    '21':+      nats: '3.044522437723422996500597980365705434284575287404610640194084483575074155970678234661274253401603616'+      bits: '4.392317422778760288895708261179647317400841033658621844133044378611419076656551549020141474088299027'+    '22':+      nats: '3.091042453358315853479175699423305867897206988297672429339247718623967245882931428680918034999053479'+      bits: '4.459431618637297256199363046725792958703231525681768071312801645726330619720018352709491299286900489'+    '23':+      nats: '3.135494215929149690806752831810196118442380314840435741998635377482993245984798298198401092152994814'+      bits: '4.523561956057012872294148244162668844498825125442555059494443732014778145562764696110754525862088214'+    '24':+      nats: '3.178053830347945619646941601297055408873990960903515214096734362117675159127693113691205735802988151'+      bits: '4.584962500721156181453738943947816508759814407692481060455752654541098227794358562522280474918088242'+    '25':+      nats: '3.218875824868200749201518666452375279051202708537035443825295782948357975415315529260267756186359222'+      bits: '4.643856189774724695740638858978780351729662786049161224109512791631869553217250431700279486718740310'+    '26':+      nats: '3.258096538021482045470719563023495172880768079120462370539725520156858289294104895005437990436908176'+      bits: '4.700439718141092160396812654256694733628436401791037369538463525842855186633025300147376530281154896'+    '27':+      nats: '3.295836866004329074185735710767577113942471673468248355204083000912482879655826900620847264441196266'+      bits: '4.754887502163468544361216831843449526279443223077443181367257963623294683383075687566841424754264726'+    '28':+      nats: '3.332204510175203923939816986359532865788084998302371696700750168924367106691458698999385152580708902'+      bits: '4.807354922057604107441969317231830808641026625966140783677291724070320848862192986497860999170210785'+    '29':+      nats: '3.367295829986474027183272032361911605494512913922744078921670351642780781137852333293367114817856423'+      bits: '4.857980995127572120719773324627984762476860500075937048051943442788096336378103434971079945416980185'+    '30':+      nats: '3.401197381662155375413236691606889912248592046451522427768022234605066902895961447109612959903330387'+      bits: '4.906890595608518529324058373437206684624645800717061672510509050357033004402983778372420218277458397'+  hypergeometric:+    10,2,2:+      nats: '0.7474814710871595424353002205959008132895071687690667637966026623611793342910689123605201166088516715'+      bits: '1.078387811493832155036127519996328468007821418919499578678169076587709370683088927073809404822971040'+    10,2,5:+      nats: '0.9950269901795212593824099014046464681916469702250978339087265011256030946130993020167734320217338660'+      bits: '1.435520504282666614090633760401527364261389152593528447547751755851121579695036449572261092414304176'+    10,2,8:+      nats: '0.7474814710871595424353002205959008132895071687690667637966026623611793342910689123605201166088516715'+      bits: '1.078387811493832155036127519996328468007821418919499578678169076587709370683088927073809404822971040'+    10,5,2:+      nats: '0.9950269901795212593824099014046464681916469702250978339087265011256030946130993020167734320217338660'+      bits: '1.435520504282666614090633760401527364261389152593528447547751755851121579695036449572261092414304176'+    10,5,5:+      nats: '1.235866276237501437415733987203138056856872737446438208882059671380121793756824376566409242187707681'+      bits: '1.782978147929753081717861035505657921666942359953125499718248628755821319116336270569922458213986168'+    10,5,8:+      nats: '0.9950269901795212593824099014046464681916469702250978339087265011256030946130993020167734320217338660'+      bits: '1.435520504282666614090633760401527364261389152593528447547751755851121579695036449572261092414304176'+    20,2,2:+      nats: '0.5172152102791041129553230505917346437690991633144458813666523863532445342889461111533164853628601624'+      bits: '0.7461838189420059150723366394158745644797403294939282785539353884294489532850532969532924198940674680'+    20,2,5:+      nats: '0.8496383026252035448134413853769020153464725539334277158947647840411483310544806008554235405745220360'+      bits: '1.225768965746697493617973394319184582304366813935560756306483475438373216736192802460358732329646111'+    20,2,10:+      nats: '1.020094373854497117875383236186510608448624328780448681460504608529814766433606037891406505153877138'+      bits: '1.471685094398615244063493355225013427097456344992562161489360955080401215237848766597316160376424419'+    20,5,2:+      nats: '0.8496383026252035448134413853769020153464725539334277158947647840411483310544806008554235405745220360'+      bits: '1.225768965746697493617973394319184582304366813935560756306483475438373216736192802460358732329646111'+    20,5,5:+      nats: '1.247673279616172846414580108421158201190032728727577037910456851035242831454395745575001619265833728'+      bits: '1.800012053151921559224521579111895619063902103155378256424309736929313564440780981634547028902144749'+    20,5,10:+      nats: '1.410178992589031228652819447306708741437597768533311273935985517608637596611516046058897830503149884'+      bits: '2.034458239373989634221859674560713137552345239847121861610699015808455486701181968646063973142589464'+    20,10,2:+      nats: '1.020094373854497117875383236186510608448624328780448681460504608529814766433606037891406505153877138'+      bits: '1.471685094398615244063493355225013427097456344992562161489360955080401215237848766597316160376424419'+    20,10,5:+      nats: '1.410178992589031228652819447306708741437597768533311273935985517608637596611516046058897830503149884'+      bits: '2.034458239373989634221859674560713137552345239847121861610699015808455486701181968646063973142589464'+    20,10,10:+      nats: '1.555951815188116695579963730964488847021215656874658798345139717291614830311643423806768088940376957'+      bits: '2.244763967634076850971891622394606806258417704630886357073461114387040892888987441338589618719612136'+  logarithmic:+    1/2:+      nats: '0.8829244358028678611733744961944307354433879906766004350294190823427973390492067733167179077816810386'+      bits: '1.273790705012483395982892455866812915659499706530622071372273071913531835757766350845756697418032690'+    1/3:+      nats: '0.5957444426529752089282563657246481688718219798805927581642526774231175248251386656391035900374057623'+      bits: '0.8594775530526067848202872635752976425515522792528938072459351654796377067686578111829621668089720239'+    1/4:+      nats: '0.4616944452661272704618622200724521231693208414645724404058230192556863601350996894431734800131608207'+      bits: '0.6660842865914227603269938800400590287199881480149275216994252109417654285050799830833911066724897286'+    1/5:+      nats: '0.3815083617781486930406096444585622895676310367784354881304725093927229804815598207217323097148839578'+      bits: '0.5504002215950076730547759683670634669483144108551243294646871197679114329221134109077580761649880490'+    1/10:+      nats: '0.2146256733097977327700470387913745065872989396977681389797236452424608948005024455284672061437511875'+      bits: '0.3096393945314999423229881594359854230011559336076745152315600223435417085671692483856979602040546990'+    11/100:+      nats: '0.2321102775566973302810445104221910373527786712051397228330649448855862256222377561258880902086242395'+      bits: '0.3348643463704081003977642332066198415830972653492620874682251106165436145652340882829227768315821531'+    2/5:+      nats: '0.7061788655876288926642071811479487931411471438884746933673414058336740053652418585655866928108773796'+      bits: '1.018800747363866059365688926175843546945898479684331941292099793730973474069285541837774468063067580'+    3/10:+      nats: '0.5418157739122489522576475826452125732573836606703275355486977707221429474931634990483766019934520831'+      bits: '0.7816749300986173552056923867685365120957508368139526365608441035687344885581751659307654932882836696'+    1/20:+      nats: '0.1215266341136756774640045442685343357925434586528077911320573803791387894668658499703513042745031811'+      bits: '0.1753258723717274267926351251017078948954670421256346383536396957327758234461123596411401495288593893'+    3/20:+      nats: '0.2998607695400497698950501978394255172511096795852178088338520653467749202434928694341638823755478706'+      bits: '0.4326076451725781358046219991911120744333990471931319097580884737332961180973016476347827867314325003'+    7/20:+      nats: '0.6229757674728398294335345496568122110770601591313233008196115178482120717991974719202367211451530801'+      bits: '0.8987640503270620175935261335766212399859263942303274954936728760265860165539854371128569138960853759'+    9/20:+      nats: '0.7924483826286087192660174347592624254320690699017577723037162754465516829393168016586726418769405844'+      bits: '1.143261351778773575764225147757404249353306578089839728317617354113597897445392889525408320750290765'+    1/25:+      nats: '0.1011257352290939186657447970895655778158553412931118575870796673141407289029510080745294387913756845'+      bits: '0.1458935967212641383142663401204175182661646340422791098621239305303175441544994005299146984173598477'+    2/25:+      nats: '0.1787235278158125545353813102444569283272681380654577542276464685941744626936568587719432086190205369'+      bits: '0.2578435472700534822604551264881641995133663928847983665734440148556016975431534330553938092626756777'+    3/25:+      nats: '0.2493433856958248547222929917538728197049682088645256162017254175415838937683238461568505547189453378'+      bits: '0.3597264660218306123673930437157843504008719218367795724085641044419399625374825792479043813729349940'+    4/25:+      nats: '0.3163927268536472707174210219308378328049737994583297993976164887070407346959620761146057981931019163'+      bits: '0.4564582180050932799012767693686786764606310649058566185486802613180785039703707464425250168217043172'+    6/25:+      nats: 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