投掷硬币问题

BeerRabbit

假设你现在有n个硬币，现在同时扔出去，将正面的硬币剔除，收集反面的硬币，继续投掷，如此进行，直到全部被剔除。 求：需要投掷的次数的数学期望。 PS：自己用matlab模拟了n从1到50的结果如下： 1 2.0014 2 2.6651 3 3.1391 4 3.507 5 3.7955 6 4.0384 7 4.2415 8 4.4211 9 4.5819 10 4.7218 11 4.8563 12 4.9744 13 5.0844 14 5.191 15 5.288 16 5.3824 17 5.4596 18 5.5472 19 5.619 20 5.6931 21 5.7609 22 5.827 23 5.885 24 5.9461 25 6.0024 26 6.0627 27 6.1132 28 6.1628 29 6.2124 30 6.2642 31 6.3094 32 6.353 33 6.3999 34 6.4427 35 6.4845 36 6.5248 37 6.5624 38 6.5991 39 6.6346 40 6.6722 41 6.7079 42 6.74 43 6.7759 44 6.8096 45 6.8385 46 6.8753 47 6.9027 48 6.9336 49 6.9637 50 6.9882 图像：

056254628

可以用递推公式 a(0)=0 a(n)=1+0.5^n*(ΣC(n,i)a(i)) i=0 to n

056254628

利用2楼的递推公式，得E(n) E(1)=2/1=2 E(2)=8/3=2.66666666666667 E(3)=22/7=3.14285714285714 E(4)=368/105=3.5047619047619 E(5)=2470/651=3.79416282642089 E(6)=7880/1953=4.03481822836662 E(7)=150266/35433=4.2408489261423 E(8)=13315424/3011805=4.42107772581558 E(9)=2350261538/513010785=4.58131019214343 E(10)=1777792792/376207909=4.72555932363453 E(11)=340013628538/70008871793=4.85672201008097 E(12)=203832594062416/40955189998905=4.97696614440968 E(13)=131294440969788022/25804920098540835=5.08796153866844 E(14)=822860039794822168/158515937748179415=5.19102401615936 E(15)=177175812995012739374/33510269239965128331=5.28720947379643 E(16)=231553634961214157747264/43060695973355189905335=5.37737790175276 E(17)=1813465925343969651214825522/332000498936684593887186105=5.46223855431559 E(18)=14983458468103810854318443432/2703432634198717407367086855=5.54238277609045 E(19)=419118293202270652551058824971246/74598662394007523860856203470915=5.61830842205474 E(20)=957245448296134815166548035810677744/168219983698486966306230738826913325=5.6904383608304 E(21)=15997819920041381172996946639598567853322/2777816590813115274614788190248819735725=5.75913470059538 E(22)=1004629398962208518335718632944853920778216/172477157411396157505627303085449443590925=5.82470985746794 E(23)=370356305063039431249700901265882284780644902994/62906221304406073331513379741466208724040808325=5.8874352549467 E(24)=19926952553863277281649067043965751602546940593375904/3350448252893971871709734118410231742851137492197825=5.94754822333138 E(25)=4355647554571850124152619507048246830952512482053380184314/725305707876137609582097583900270649738762814538483595565=6.00525751731114 E(26)=923476084609715412475893251255716866736272968716013010867368/152369991400748600905285269356279934187427788192661438422155=6.06074776352044 E(27)=4797960683843138105983901203020953803581431625690850487123547946/784726369242086769960382607712008209291514128692546577091136585=6.11418307311013 E(28)=269542159343895657509952303227407839058359990344908292442274298664656/43716321304107411867722954693028265331420960595333077263170128013765=6.16570999807732 E(29)=5030240603353911185019649911663910424320080679721904259748482335225032852254/809311077382926008664711523505441729559644415148674880645225288784021934135=6.21545996827355 E(30)=6152272295229061448667491550002707202162123303490067362560058343899776476596616/982233877583744532516071552361104445775555105185441746809755092154207954061845=6.26355131464555 E(31)=429357150728754480560662348879788829530404391182876716173286881137001622838684482265362/68042941601306169567868497528302452779419445830100033991776241305681922051133527875765=6.31009096056647 E(32)=1889321865157655935212321410354950581504203995260966246373219791333722288719259466941179520/297288684248320162331293181500823856520320814757817728514602635096698408364342534426267387=6.35517584510394 E(33)=7153419320484531952473738619667018781510703214740098858786348849530426741122690054452173677515782/1117914936088219259909080914141121322102112065273084106452925240261292990472862834134307266864067=6.39889412830962 E(34)=24003398681250390133286948336914405521560656088511043048333946198412852958186847249633454369530312/3726468411736506270289742444475450651099670271141093896012417538128950335450511183807279987530171=6.44132621804917 E(35)=8398317377316604212780871552525168440178841328095156085843678866116423181775506919057170701564984069705974742/1295527687900007543723673530264953456458053115692465187399843086483116374812796357616951808724924661663152935=6.48254564974208 E(36)=227278160621192938936250648813873556860714544804436068433294929205519796977548453870540820210476226481493553310585552/34844612449390571196572789992868018566311943869754452000125609441967557216925831419785072352754845519291109402535995=6.52261984406625 E(37)=36925499797732758842174956307878778802577836896431020231866023597051742995077975584827964243403972376601741431973438536612702/5627505369150197481776592806691471240398720777422733034830042916862629108141715156476886519283950307867971349094142451477895=6.56161076276482 E(38)=7856979944159715313072278959756522767448939001702324609812732453456583520639998243151177036534749502331364768108184301398176695432/1190528083108542480930401149449678764888075668535198376179898114549066524684311737657974354300067535581073493067369010352395861545=6.5995754788452 E(39)=5827397172205445293639704236120962448643291034117615361161427118598995698536151445281360081693548222071543304266716417554573961101704974/878074079275788822851946670115160530710797508410879295828282022070134365209108328595157266396657449334554261048260557229179594789888715=6.63656667443335 E(40)=337902510094510736525670256499889776984957775193708928611681316621640961772682147508153168262579843878556807472085008246994041731992405139264224/50640055610362284907479412593013957609612475543732501865995286282092940524732915113032832150952433303798339686296105067473579858129842650970525=6.67263307715182 E(41)=18218873878277805235602500862421320459916017065238677505841120887774607512406872264623551152821004032367366109642877686753095472495680641019296704765876214/2716064876819086686043716358775578040533481363455683295786613092265079732357385200649463114774077115830824867096278659181631709714757739044014051103503275=6.70781984398502 E(42)=99233639167261065056160413546786758551878469701526039592821531883710784980903797092110584834022369397300973746593333441093682846967650625345156963589009581816/14718355567482630751670898948204857401650935508566347779867656346984467069644670402319440618960723890687239954794734054105262234944272187879512142929884247225=6.74216890007051 E(43)=20400276346676396531883380183447924375883847222405738938192420384140449925410956508553064359706844939827366188919297442941154535671357342865072696566422128240003559942434/3010791272220826001475439852194599109289424432298005036762612048084574102267815302994149093410788570117850976655391030844269245290075033332770094885601871531861195421525=6.77571923862683 E(44)=1563255147218453018818614077479104584228295421285565580335292508391581055316100139626942882755105424161934261292988540363398146258344939180512686971864647192199350166220298224/229603216127857665493062631804710327583421442609266982103974682297115810545680164807724991149787955519784598920737997775004958488479784776050986778711897414098964932045080275=6.80850718723353 E(45)=23102570759834218693908461758875860792345792168661782946449239780543695897351515877774381924386247203751090409572609958538076886235675307847678789053988439853581503857461169803908774/3377289040399745215434842079309749143613493603907376873110050659806710062226751053076545556545098136637791933613475107130011710602044877899064992815886968631258473035999708629321275=6.84056664486725 E(46)=2821550153361745541848913038990129594470771701398426083340199669101915017912804285360809519809664415062167351517354285230563307141400026159024122181087028175620914491169061145942619867864/410590684636212555244980509856180803679107898075070388466127955874065308527331651081121561515133610650478422528078258042706962400894334509391270874207025623810036351247021446453653787775=6.87192929343165 E(47)=8486630560567079345224854302687910433640039177826588989723769285021826304402891341324576244839811170578419738509837138196499706031354129978915350027923203357198792691079783827228769950208598334578774/1229478759527546916148325110290453557644382129466634340323990585506269022275439144662568220722958711400133184178869998850656123488367115346936618514509210301404623384994473456075832596110128694653775=6.90262478696927 E(48)=7233565080841646748303518698632566039487692375493151907582665590007438523009999668673731375292937136786729265213485034816234533465039794821363895242440031237039887089514078549370060662247674153426883767232/1043400837709331273030024549823324283055577827318247632818975582361651724561118256435322508221201079305857227152950561134605871166773217540523802112193784849917937849545714881919123560186649047104010115075=6.93268092128632 E(49)=4600033011954980440233977994246560790358679502835095869284656466018690600628699170359441798623381140662058001058784734357587654689905019075237019939931191366931240273084532361921590080824495355486666392531544669140618/660722669278093271259660917754149872396525097069829213659259606332097737822745014414258770843632646186686216618553269326306811821193444503083285770142199662366581389384091763169545377530365150236776802111848646869925=6.96212378633986 E(50)=10332746687276762385115122228406199608676601608975702653299579220748965954694430427169775808642837349873661051181533428302893625974702475381462183345692300813338871306635904226681345426995007475142461048909987112232071510664/1478011635858195935882207857833342157685850053496538711411487419351783286214990897483152011395668340205591209836715482169367803646522894241195093719605989269566805111275694555541625200720156193276990756161085595169170541835=6.99097790341626

KeyTo9_Fans

https://oeis.org/A158466

056254628

好像极限趋向于7，其实不然。 E(51)=7.01926634885887 E(52)=7.04701086587346 E(53)=7.07423196574364 E(54)=7.1009490196214 E(55)=7.12718034191357 E(56)=7.15294326616076 E(57)=7.17825421419844 E(58)=7.2031287592958 E(59)=7.22758168388782 E(60)=7.25162703244486 E(61)=7.27527815996257 E(62)=7.29854777650031 E(63)=7.32144798814986 E(64)=7.34399033477339 E(65)=7.3661858248141 E(66)=7.3880449674493 E(67)=7.40957780232858 E(68)=7.43079392711343 E(69)=7.45170252301276 E(70)=7.4723123784892 E(71)=7.49263191129263 E(72)=7.51266918896322 E(73)=7.53243194793091 E(74)=7.55192761132675 E(75)=7.57116330561051 E(76)=7.5901458761086 E(77)=7.60888190154818 E(78)=7.62737770766507 E(79)=7.64563937995574 E(80)=7.66367277563874 E(81)=7.68148353488305 E(82)=7.69907709135759 E(83)=7.71645868215079 E(84)=7.73363335710497 E(85)=7.75060598760643 E(86)=7.76738127486935 E(87)=7.78396375774764 E(88)=7.80035782010708 E(89)=7.81656769778651 E(90)=7.83259748517607 E(91)=7.84845114143605 E(92)=7.86413249638092 E(93)=7.87964525604884 E(94)=7.89499300797686 E(95)=7.91017922620003 E(96)=7.92520727599137 E(97)=7.9400804183588 E(98)=7.95480181431312 E(99)=7.96937452892116 E(100)=7.98380153515692 E(200)=8.98020421977786 E(300)=9.56396933583119 E(400)=9.97840328257279 E(500)=10.2999737329228 E(600)=10.562768015093 E(700)=10.7849866359562 E(800)=10.9775022445484 E(900)=11.1473282430783 E(1000)=11.2992526972793

KeyTo9_Fans

$E(n)=\log_2 n+O(1)$ 有没有更好的近似公式？

056254628

好像 E(n)-ln(n)/ln(2) 存在极限 近似公式：E(n)=ln(n)/ln(2)+4/3

056254628

以下是 E(n)-ln(n)/ln(2)-4/3的值：以下的E()是减了ln(n)/ln(2)+4/3的值 E(1)=.666666666666667 E(2)=.333333333333333 E(3)=.224561308802653 E(4)=.171428571428571 E(5)=.138901398200195 E(6)=.116522394312126 E(7)=.10016067075136 E(8)=8.77443924822491E-02 E(9)=7.80518573677807E-02 E(10)=7.02978954138316E-02 E(11)=6.39570581103412E-02 E(12)=5.86703103551883E-02 E(13)=5.41884871940157E-02 E(14)=5.03357607684278E-02 E(15)=4.69855448545793E-02 E(16)=4.40445684194245E-02 E(17)=4.14423797319152E-02 E(18)=3.91244413148058E-02 E(19)=3.70475752778217E-02 E(20)=.035176932609704 E(21)=.033483944483291 E(22)=3.19449054973077E-02 E(23)=3.05399655563512E-02 E(24)=2.92523892768931E-02 E(25)=2.80679942030785E-02 E(26)=2.69747120460114E-02 E(27)=2.59622376133298E-02 E(28)=2.50217426863835E-02 E(29)=2.41456398126401E-02 E(30)=2.33273857036998E-02 E(31)=2.25613168462645E-02 E(32)=2.18425117706054E-02 E(33)=2.11666756178367E-02 E(34)=2.05300434654932E-02 E(35)=1.99292994637755E-02 E(36)=.01936150929061 E(37)=1.88240638025379E-02 E(38)=1.83146320682808E-02 E(39)=1.78311222377641E-02 E(40)=1.73716489311192E-02 E(41)=1.69345060336029E-02 E(42)=1.65181439584161E-02 E(43)=1.61211505913976E-02 E(44)=1.57422352629026E-02 E(45)=1.53802152042413E-02 E(46)=1.50340040413038E-02 E(47)=1.47026019583025E-02 E(48)=1.43850872318332E-02 E(49)=1.40806088913238E-02 E(50)=1.37883803081993E-02 E(51)=1.35076735540452E-02 E(52)=1.32378143990388E-02 E(53)=1.29781778471088E-02 E(54)=1.27281841245964E-02 E(55)=1.24872950555796E-02 E(56)=1.22550107698206E-02 E(57)=1.20308667003622E-02 E(58)=1.18144308348911E-02 E(59)=1.16053011926421E-02 E(60)=.011403103503005 E(61)=1.12074890663451E-02 E(62)=1.10181327801041E-02 E(63)=1.08347313166117E-02 E(64)=1.06570014400536E-02 E(65)=1.04846784523172E-02 E(66)=1.03175147575145E-02 E(67)=1.01552785374702E-02 E(68)=9.99775252975491E-03 E(69)=9.84473290126079E-03 E(70)=9.69602821090263E-03 E(71)=9.5514584546153E-03 E(72)=9.41085418757499E-03 E(73)=9.27405571755647E-03 E(74)=9.14091236447066E-03 E(75)=9.01128178129747E-03 E(76)=8.88502933167881E-03 E(77)=8.76202751994326E-03 E(78)=8.6421554694879E-03 E(79)=8.52529844530336E-03 E(80)=8.41134741804908E-03 E(81)=8.30019866509556E-03 E(82)=8.1917534061732E-03 E(83)=8.08591747053332E-03 E(84)=7.98260099287366E-03 E(85)=7.88171813539588E-03 E(86)=7.78318683391623E-03 E(87)=7.68692856558144E-03 E(88)=7.59286813644687E-03 E(89)=7.50093348678293E-03 E(90)=7.41105551306427E-03 E(91)=7.32316790402176E-03 E(92)=7.23720699057783E-03 E(93)=7.15311160747715E-03 E(94)=7.07082296589138E-03 E(95)=6.99028453574926E-03 E(96)=6.91144193687833E-03 E(97)=6.8342428383424E-03 E(98)=6.75863686457307E-03 E(99)=6.68457550822096E-03 E(100)=6.61201204886049E-03 E(200)=3.01469666979734E-03 E(300)=1.81731200197274E-03 E(400)=1.21375946473405E-03 E(500)=8.56114927414704E-04 E(600)=6.15991263752591E-04 E(700)=4.42190790548309E-04 E(800)=3.12721440340659E-04 E(900)=2.13718527913548E-04 E(1000)=1.35079283888664E-04

056254628

是不是E(n)-ln(n)/ln(2)的极限就等于4/3?

kastin

056254628 发表于 2012-3-30 16:59
可以用递推公式
a(0)=0
a(n)=1+0.5^n*(ΣC(n,i)a(i))    i=0 to n

能说一说过程吗？