论坛 › 趣题妙解 › 训练营成员的能力值分布

KeyTo9_Fans 发表于 2021-10-25 15:00:53

精确计算了$3$个新人闯营的情况，结果如下：

=====

第$1$人闯营后，除去能力值为$0$的初始成员，

营中有$1$人的概率为$1$，他的能力值的概率密度函数为$f(x)=1$，$0<x<1$；

加上能力值为$0$的初始成员，营中人数的期望值为$2$。

=====

第$2$人闯营后，除去能力值为$0$的初始成员，

营中有$1$人的概率为$1/4$，他的能力值的概率密度函数为$f(x)=2x$，$0<x<1$；

营中有$2$人的概率为$3/4$，他们的能力值的概率密度函数为$f(x,y)=2$，$0<x<y<1$；

加上能力值为$0$的初始成员，营中人数的期望值为$11/4$。

=====

第$3$人闯营后，除去能力值为$0$的初始成员，

营中有$1$人的概率为$1/12$，他的能力值的概率密度函数为$f(x)=3x^2$，$0<x<1$；

营中有$2$人的概率为$5/12$，他们的能力值的概率密度函数为$f(x,y)=(12x+9y)/5$，$0<x<y<1$；

营中有$3$人的概率为$1/2$，他们的能力值的概率密度函数为$f(x,y,z)=6$，$0<x<y<z<1$；

加上能力值为$0$的初始成员，营中人数的期望值为$41/12$。

=====

各种多重积分、条件概率、边缘分布的处理太容易出错了，我花了$2$天的时间纠错，才得到上面的结果。

#####

依次类推，$4$人的情况初步计算结果如下：

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第$4$人闯营后，除去能力值为$0$的初始成员，

营中有$1$人的概率为$1/32$，他的能力值的概率密度函数为$f(x)=4x^3$，$0<x<1$；

营中有$2$人的概率为$31/144$，他们的能力值的概率密度函数为$f(x,y)=(84x^2+84xy+54y^2)/31$，$0<x<y<1$；

营中有$3$人的概率为$127/288$，他们的能力值的概率密度函数为$f(x,y,z)=(696x+528y+432z)/127$，$0<x<y<z<1$；

营中有$4$人的概率为$5/16$，他们的能力值的概率密度函数为$f(x,y,z,w)=24$，$0<x<y<z<w<1$；

加上能力值为$0$的初始成员，营中人数的期望值为$581/144$。

=====

这个结果有待验证。

mathe 发表于 2021-10-26 08:36:05

也就是说Fans发现n个人闯营以后，还保留h个人的密度分布函数是h个变量的n-h次齐次函数，这个结论很有意思。
假设n个人闯营以后，留下h人的概率为$p_{n,h}$, 这h个人的密度分布函数是n-h次齐次函数$f_{n,h}(x_1,x_2,...,x_h), 0\lt x_1\lt x_2\lt\cdots\lt x_h$
那么第n+1个人闯营以后，有$1/{h+1}$的概率选择和$x_t$做比较 $0\le t\le h$.
   其中闯营失败的密度分布为$\int_0^{x_t} x_{h+1} f_{n,h}(x_1,x_2,...,x_h)d x_{h+1} =x_t f_{n,h}(x_1,x_2,...,x_h)$,是需要添加到n+1个人留h个人的密度分布函数中，正好是n-h+1次齐次函数。并且计算这个函数多所有变量的累次积分可以得到这部分总的概率。
   闯营成功，那么落在$(x_s, x_{s+1})$的密度分布为$f_{n,h}(x_1,x_2,...,x_s,x_{s+2},x_{s+3},...,x_{h+1})$, 需要将这个函数添加到n+1个人留h+1个人的密度函数中，这正好是一个(n+1)-(h+1)次齐次函数

所以逻辑代码如下

$f_{1,1}(x_1)=1$

for(n=2, any){
f_{n,1..n}=0
for(h=1,n-1){
   for(t=0,h){
      if(t>0) $f_{n,h}(x_1,x_2,...,x_h)+=\frac{x_t f_{n-1,h}(x_1,x_2,...,x_h)}{h+1}$
      for(s=t,h)$f_{n,h+1}(x_1,x_2,...,x_{h+1}) += \frac{f_{n-1,h}(x_1,x_2,...,x_s,x_{s+2},...,x_{h+1})}{h+1}$
   }
}
}
比如这样经过一轮后可以得出$f_{2,1}(x_1)=\frac{x_1}2, f_{2,2}(x_1,x_2)=\frac3 2$, 正好对应Fans的n=2时的密度函数和前面的概率的乘积

mathe 发表于 2021-10-26 11:55:57

Level 1:
        1

Level 2:
        +1/2*A
        3/2
       
1/4 3/4
Average 11/4
Level 3:
        +1/4*A^2
        +1*A+3/4*B
        3
       
1/12 5/12 1/2
Average 41/12
Level 4:
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        +7/12*A^2+7/12*AB+3/8*B^2
        +29/12*A+11/6*B+3/2*C
        15/2
       
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Average 581/144
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        45/2
       
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Average 79823/17280
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        315
       
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        +21321/56*A+4167/14*B+2487/10*C+4311/20*D+765/4*E+345/2*F+315/2*G
        2835/2
       
1/1024 1133/68040 1569971/19906560 368368771/1935360000 540359959/1935360000 233002181/903168000 176651/1254400 9/256
Average 95151902939/15240960000
Level 9:
        +1/256*A^8
        +2123/93312*A^7+617/15552*A^6B+445/7776*A^5B^2+379/5832*A^4B^3+445/7776*A^3B^4+617/15552*A^2B^5+2123/93312*AB^6+3/256*B^7
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        +5074911/15680*A^2+1177341/3136*AB+107439/490*B^2+4977001/15680*AC+963259/3920*BC+8227/50*C^2+4341053/15680*AD+1684409/7840*BD+28563/160*CD+52299/400*D^2+19331701/78400*AE+7514061/39200*BE+127571/800*CE+13771/100*DE+1725/16*E^2+7491011/33600*AF+10204267/58800*BF+57801/400*CF+37463/300*DF+5305/48*EF+365/4*F^2+9591161/47040*AG+2335451/14700*BG+26477/200*CG+68681/600*DG+4865/48*EG+365/4*FG+315/4*G^2
        +116685/64*A+641925/448*B+134595/112*C+16713/16*D+29727/32*E+26865/32*F+12285/16*G+2835/4*H
        14175/2
       
1/2304 2133889/235146240 738921023/15049359360 20159805067/149299200000 362925784139/1567641600000 1353647390737/5120962560000 17026982389/85349376000 1652579/18063360 5/256
Average 3112703236301951/460886630400000
Level 10:
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        +5800951/3584*A^2+47928733/25088*AB+27700299/25088*B^2+725873/448*AC+31639677/25088*BC+40806/49*C^2+177722663/125440*AD+138655257/125440*BD+3613941/3920*CD+212607/320*D^2+396554453/313600*AE+619798257/627200*BE+129385917/156800*CE+2286909/3200*DE+351783/640*E^2+102627457/89600*AF+2294349/2560*BF+29361933/39200*CF+2077293/3200*DF+368553/640*EF+59715/128*F^2+32901383/31360*AG+515390343/627200*BG+10775997/15680*CG+953481/1600*DG+169239/320*EG+61065/128*FG+12915/32*G^2+121610249/125440*AH+95325147/125440*BH+6231951/9800*CH+110331/200*DH+313443/640*EH+56565/128*FH+12915/32*GH+2835/8*H^2
        +153955/16*A+486275/64*B+2864475/448*C+624075/112*D+79455/16*E+143865/32*F+131775/32*G+15225/4*H+14175/4*I
        155925/4
       
1/5120 35303591/7054387200 493315123/16124313600 177040148897/1881169920000 4913011142447/26873856000000 15756307796485993/64524128256000000 20497966660523/89616844800000 166085069503/1137991680000 3749981/65028096 11/1024
Average 1403936289503846939/193572384768000000
Level 11:
        +1/1024*A^10
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        467775/2
       
1/11264 6496405/2327947776 569783399443/29797731532800 62996436704843/969978240000000 161010876007913/1149603840000000 1315083796508393489/6210447344640000000 192607568465219557/828059645952000000 729523557898043/3943141171200000 4609294575701/45064470528000 42308191/1192181760 3/512
Average 865440371749038462403/111788052203520000000
Level 12:
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        6081075/4
       
1/24576 146408989/93117911040 8583072000373/715145556787200 498197701343978243/11174149324800000000 24496866073770929/232794777600000000 441238531261177743389/2504052369358848000000 2745921459415330190117/12520261846794240000000 2038875590064298429/9936715751424000000 3225051882695327/22712493146112000 34338052012597/495709175808000 561745903/26227998720 13/4096
Average 8492672176875167143532237/1032921602360524800000000

Level 13:
1/53248 721853003/807021895680 281503678355437/37187568952934400 88715784400749967651/2905278824448000000000 37798534825398160559/484213137408000000000 1624568664901010792829401/11393438280582758400000000 597201188730712614008399/3038250208155402240000000 271840207150663884092677/1302107232066600960000000 7206364035383776819/42276208833331200000 62218993549362983401/595445861980569600000 673746715476949/14768002529280000 7434422179/584509685760 7/4096
Average 719162129942703833322750483827/82716361917030825984000000000

Level 14:
1/114688 52210603817/101684758855680 2498623818382213/520625965341081600 1417146972089835739427/67789839237120000000000 2123479359236610436013/36976275947520000000000 40108924042491076211600383/354462524284796928000000000 260340554695322607053464361/1531278104910322728960000000 919767658082469049944284141/4593834314730968186880000000 6103543220301401334681989/32813102248078344192000000 93543322656237650447677397/693241890352258351104000000 56767708423524073013/764155522875064320000 3030035245383284053/103210616076632064000 7549263763/1013150121984 15/16384
Average 91008379133205771959239329083567887/9935889393473742817198080000000000

Level 15:
1/245760 38858238427/130737547100160 286654638384296387/93712673761394688000 524317697587972030903487/36606513188044800000000000 27470205268281938382581/653687735500800000000000 16481168623071937881494529653/186092825249518387200000000000 230461070660662046085895546481/1607842010155838865408000000000 332937408216669435208270547753/1808822261425318723584000000000 78516278535000936213639608359/413445088325787136819200000000 580641756417032863205816631661/3695512538570115671654400000000 73963260275885859687239/720489492996489216000000 5964476097641414388649279/116228055029297283072000000 1775418549993086407/95838429214015488000 38251327319/8865063567360 1/2048
Average 430482610611138238966092369119663819653/44756213772902474520068751360000000000

Level 16:
1/524288 3091326263/17827847331840 3526412638168580179/1799283336218778009600 57571002542002385913017683/5857042110087168000000000000 76937532337595104977215977/2510160904323072000000000000 6952503351845960447728646209399547/101294046639817848520704000000000000 3203221648348252829113850434898989/27011745770618092938854400000000000 6835521222667500019563484155780187/41675264903239343391375360000000000 370126232255892789346228568939/2003618504963429970739200000000 144927937824466983151671672077843987/852297534975441205399922933760000000 33558129078692645831227750026007/264229146507763270523289600000000 2500003356918612233697998617073/33093484966208449523220480000000 238954724342083526719935457/6907267270312524251136000000 23502514183211758601/2044553156565663744000 2631526038067/1063807628083200 17/65536
Average 6497374168950664548739248365120838691744731463/645133967808125428722079009603584000000000000

Level 17:
1/1114112 1863464628559/18288315160068096 66188299948170023399/52436257226947244851200 40385906698116373280509362629/5974182952288911360000000000000 22218887681869543934889582691/995697158714818560000000000000 424449809305769911205898210910751441/8035994366758882649309184000000000000 24876635804464439104365649844381920603/257151819736284244777893888000000000000 254889920303821362019123288140992935447/1785368348454773470886520422400000000000 12628611959473131391838847896596033/72872177487949937587204915200000000 58363586040687553029368958978065141740241/334697241984855761360549736087552000000000 19281842844302909706866894892029978789/132816365867006254508154657177600000000 812310853734815176311723010378115121977/8210753734908150725124253600972800000000 9146780377555490265092837488615763/169258992680027844261477089280000000 5005907400871473007812936173/219190614711250769569382400000 1603556631633520054331/228095461529356861440000 45197664707939/32150630537625600 9/65536
Average 166290574652981280453599725306444126898090108578925147/15808672411474853405611610649250832056320000000000000

Level 18:
1/2359296 694495381956869/11521638550842900480 899692116487219245181/1101161401765892141875200 1879518068144214766515427683761/403257349279501516800000000000000 3080666257483704245374332616451/189768164366824243200000000000000 2456945008113422743031710861837506101207/60752117412697152828777431040000000000000 91080191659578309752096300063073263104753/1166440654323785334312526675968000000000000 10767379843382292204897474558723786727049/88218200747177042090863362048000000000000 1571338083053592507378366368657973023/9937115111993173307346124800000000000 150567368160565984436975415678164136948210187/877103654730901421683087690747084800000000000 51844898966907878597657386291328834640250673/331350269565007203746944238726676480000000000 2113923747234756480284869396073440195693418041/17752963295469007171834656165879349248000000000 122974361191777525381361831238699894921193/1646842606258720516867778865109401600000000 53638702451676643940708540885432209/1421775538512233891796407549952000000 438097493609107713150601502789/29590732986018853891866624000000 2370493538295982547528981/558377689823865596805120000 866632158911641/1093121438279270400 19/262144
Average 117953595674786257235074513743736834110025844842732606522003/10760647137042703216131711136732056364095897600000000000000

Level 19:
1/4980736 100698513393601/2806552980333527040 33355049570040529612889/62766199900655852086886400 89776097358043250445541654564243/27861416859311013888000000000000000 1551183296331865733947880366074273/131346679479609065472000000000000000 5609850967158477292836887020953886006687511/181800711357496229840116462387200000000000000 129324087609038853856484788220979457810492219/2068488093667512659514213972049920000000000000 1023106037973564506400479651375220528853958311/9942381232639231178807881223897088000000000000 260725704328741809231464741335068596365103039/1846442228918714361778606513009459200000000000 3025380262497952547074280602709117492123858052197/18478118114946846430881944998506984898560000000000 167452455164873426620947884458071687572764193621/1038783095086297583746670188408130764800000000000 1532878826573189559170359872499488426931267748643817/11395471800932720349787912239636502837985280000000000 16627125840588929095780522541961477652316281097/176182309847444655995484110074775863296000000000 114411161101793953892359613218594332317111/2086000634594379321365853229138575360000000 435334952633006554677409579961743967/16883584519832777465082339655680000000 3077314432152368625761405346703/325779879350836143799979212800000 1625684306805583731334567/641265755779941645397524480 16601116665837019/37384753189151047680 5/131072
Average 385505462910834226878588650249250257267223622679349180575265679637989/33817481022899363098547713491454933373339377185325056000000000000000

Level 20:
1/10485760 6717316440139801/312730189237164441600 83030977749805266867149/239109332954879436521472000 268497750485095260966132636325091/120186504098988687360000000000000000 948419078627571458955250878853195051/110331210762871614996480000000000000000 8967245050904920961964072101144381134228053239/381781493850742082664244571013120000000000000000 1296764465116125138579904485960474278406543756647/26062949980210659509879096047828992000000000000000 2117572418960017128850345572609579897601876060283/24675182513731910107405014310217318400000000000000 192266800940925669522355920814346793146559405323/1551011472291720063894029470927945728000000000000 20403860372598931451558958303139160348546663084085410263/133963513545962337109058580939945100978421760000000000000 1357788286525232148411215324611713203972012926233343/8469137469350637947487558124315701411840000000000000 377279426904890680015276180487946670390464098343589748647/2607283948053406416031474320428831849331032064000000000000 635809025280855033102282232590081611428130258587721/5702876714812645612645740140377485931551129600000000 191040955726512532575574891238006446244657588669/2630989160388506862865896043783319558553600000000 114883914604532350588950213778786800005024059/2920400888432131049912194520794005504000000000 5509482737976433579448983739963688824627933/318524354864403593034046213356886425600000000 38907695737165436536209635128648697/6547451619664360187837804445696000000 1304478692866158075393944821/870289239987063661610926080000 3344738931830159/13529720201787998208 21/1048576
Average 15526225346871553628852438219649810173292958697066691663132713538499126698241/1312076330012026893013409084303722010768184893962902369730560000000000000000

mathe 发表于 2021-10-27 07:20:26

迭代式可以另外写成
$f_{n,h}(x_1,x_2,...,x_h) = f_{n-1,h} (x_1,x_2,...,x_h) \frac{\sum_{t=1}^h x_t}{h+1} +\sum_{t=0}^{h-1} \frac{(t+1)f_{n-1,h-1}(x_1,x_2,...,x_t, x_{t+2},...,x_h)}{h}$, 而且$n\lt h$时，$f_{n,h}=0$, 而且$f_{n,0}=0$
所以马上迭代可以得出$f_{n,1}(x_1)=f_{n-1,1}(x_1) \frac{x_1}2$, 所以$f_{n,1}(x_1)=(\frac{x_1}2)^{n-1}$, 所以n次试验后还是只有一个人的概率为$\frac1{n2^{n-1}}$
而对h比较小的情况都可以比较容易计算出来，比如
$f_{30,2}(x_1,x_2)=22876658237233/2046980738154938499072x_1^28 + 2541630947305/113721152119718805504x_2x_1^27 + 2540062277609/56860576059859402752x_2^2x_1^26 + 2532858560489/28430288029929701376x_2^3x_1^25 + 2508920094697/14215144014964850688x_2^4x_1^24 + 2447591636969/7107572007482425344x_2^5x_1^23 + 2321571839977/3553786003741212672x_2^6x_1^22 + 2108222840809/1776893001870606336x_2^7x_1^21 + 1804850022377/888446500935303168x_2^8x_1^20 + 4311895963067/1332669751402954752x_2^9x_1^19 + 1053702648937/222111625233825792x_2^10x_1^18 + 705873870089/111055812616912896x_2^11x_1^17 + 429969342257/55527906308456448x_2^12x_1^16 + 237450723259/27763953154228224x_2^13x_1^15 + 237450723259/27763953154228224x_2^14x_1^14 + 429969342257/55527906308456448x_2^15x_1^13 + 705873870089/111055812616912896x_2^16x_1^12 + 1053702648937/222111625233825792x_2^17x_1^11 + 4311895963067/1332669751402954752x_2^18x_1^10 + 1804850022377/888446500935303168x_2^19x_1^9 + 2108222840809/1776893001870606336x_2^20x_1^8 + 2321571839977/3553786003741212672x_2^21x_1^7 + 2447591636969/7107572007482425344x_2^22x_1^6 + 2508920094697/14215144014964850688x_2^23x_1^5 + 2532858560489/28430288029929701376x_2^24x_1^4 + 2540062277609/56860576059859402752x_2^25x_1^3 + 2541630947305/113721152119718805504x_2^26x_1^2 + 22876658237233/2046980738154938499072x_2^27x_1 + 3/536870912x_2^28$
而n次尝试后只有两人的概率为
$3/4, 5/12, 31/144, 481/4320, 1511/25920, 1123/36288, 1133/68040, 2133889/235146240, 35303591/7054387200, 6496405/2327947776, 146408989/93117911040, 721853003/807021895680, 52210603817/101684758855680, 38858238427/130737547100160, 3091326263/17827847331840, 1863464628559/18288315160068096, 694495381956869/11521638550842900480, 100698513393601/2806552980333527040, 6717316440139801/312730189237164441600, 20974007186069533/1622517805336347279360, 2033078119946703031/260066425369625949634560, 7747044278977776727/1631325759136744593162240, 708842463875010695/244698863870511688974336, 3472329061473680008921/1957590910964093511794688000, 33285821273155605689963/30538418211039858783997132800, 3196484217123464894063/4756945913642747233661091840, 174772565732140876609667/420614164995779755397401804800, 3010875178412763609198499/11685297726689477910452187955200, 24479043250136758346740471/152807739502862403444374765568000, 4471438961895272154536275/44786559287020762609514931290112, 192023331828123402046045157/3079075950982677429404151526195200, 3046366705306086219165459058373/78036100901704976770818816279891148800, 17717526808463051462476482525629/723607481088537057329410841868081561600, 6876444454580051974690022065079/446934032437037594232871402330285670400, 22261130072215796787893184908999/2298517881104764770340481497698612019200, 86554639773145124633026021162391/14174193600146049417099635902474774118400, 6231047111436711838508363907645047/1615858070416649633549358492882124249497600, 441196714613084942858043654346259/180914252860045939832798989155223480565760, 35463048801901396499572257554503519/22962193632236600055701410162009134071808000, 4425622377025390277820323084131729487/4518959706824162890962037519883397585331814400, 118015135445412213691088621617003692499/189796307686614841420405575835102698583936204800, 794526104871041982589696888968021343/2010157938552817285979664965741235477613117440, 19355179836206624436737101527448228060601/76948845887801845707301574888574494083030135603200, 43236778750886109779921898706470769605743/269820628437746731700927600258637836395040735232000, 5327065470711929681629778036120855980110509/52129345414172668564619212369968829991521870046822400, 4169283269768491492603053575849176213193791/63915110464333445805315729949266130685257249361756160, 105710671939984690384946588338573449863027/2536313907314819277988719442431195662113382911180800, 13356167921382951322650332140667613102830312023/501094466040374215113675322802246464572416834996168294400, 387286673623728991362785099798175926129563747/22701350016930272505904952709253086585884181591490560000, 350321476849239374763588532979225426381928359/32056078036589792994728498002262178012367850489650348032, 2112440029460789452212326173717770489941268021793/301515698708865170579828637091866015539859958429122979430400, 1275093407929901115242507621306663350129977664973/283674533104790308415341735488797020537501381007281383014400, 30247475953508143117327042222407491788991008963061/10481027486292778763556310437533447811438208919321659519795200, 12197176690593763669374559803696492418868078851639/6578182416321894178928735379713184962734491483898579098009600, 276961244362849151059334591451246766020378433766339/232329442612823262592164881365324759820213631044963452688793600, 139287127325578839653826238016200548946331347325347753/181615244282481270414903747261579583676601284154005716159011225600, 821151094570308422558530069932407670243687048516161069/1663213289744828476431223790711307766301506496989315505877260697600, 1834277800192758536846844622312233812429833357476019/5767804707774973360664706327505906634705459442929248521557114880, 62421243942586712069766019967933824310863393333458498101/304540088570518593443096494092311870312448258586664321938215665664000, 982202457416230233870709420614933972908426705461331985011/7430778161120653680011554455852409635623737509514609455292462242201600, 117877245289069226443481469669518093337489077785862464896371/1382124737968441584482149128788548192226015176769717358684397977049497600, 18558189665587058282268429142151132251173093653773105777745/337060097388432850925324110117465301071892733431582684892066087306264576, 373343050370299087689945006852959736553506673663441644539/10498134262513481652574463562628695008468419853558430800209699563110400, 159349450482186982617736948491907296107567171799692771001983773/6933807717704904361892381693845000479193221944878272374922502367443156992000, 453309208957993540815966083645652707347990368526696024971265331/30508753957901579192326479452918002108450176557464398449659010416749890764800, 508818551258497305045082999698216647152200100964748995416691/52941893685040295412739552534201143259936851317019287648980929757115842560, 117885296318111256570720934957321934864012244826271899433769804161/18954256777118126563669014598294693309922591508519245364088152471642613953331200, 4005647295127009378632013615583640608722016238112497924511802599/994810130036372261328470086675203223923422018829483719666898468973129079193600, 343476048152840959893069470444655850880547674801653110901980186289/131704210693945805901964844083738861602035654231990344625465471218399045484544000, 26427604009574282201162321439797724027679108608063443183535853279/15639301395295028101529483924647457851558798244690700156222520066630799547432960, 11493158146895708082187766534586264998390323313423581820088318428161/10492549481570664308117053760354385358591266458710669741174745281066845514550476800, 419978813784807352297842070049935442875151395905827828543201497924751/591264367277280241362666257513303259154300839743485459800934769171696627942739148800, 1149174591924583289300169303244500026067184943226386440952104908444738827/2493953101175568058067726274191113147112840942038021669440342856366216376662473729638400, 226750960329923103974764643783915005383204434649385509297742342895165413/758296551033111909547619475260811429865390826971020102194698841462700925336562958336000, 13424937287584427702145823557746436838794541710061913773042566209126188793/69156645454219806150742896143786002403723643419757033320156534341398324390694541800243200, 42398527794047998411323573825026184809953191442211282014164090055733064769/336319686314205794122560189667675085373898139578186835725392830165537114405272403281182720, 294938554514419520090268851193076123715325188485968802737456544835124643/3601347510022796180988947968765088363741255716083931411353428070743787227644936637644800, 10394716982379977739545457223756705514557517339042380019697081712451325697/195314299574777684898423829824158282856196665229184709793967411784445714565179023713894400, 141367225500872621513469182468905513335385989620205046714479093336946598473/4086180741103901565638077492373837759754640759400048533847476114964061659982034838224896000, 217304141079792476182414620742153031503329090629651538519799572440085968098079/9659338998618477326618113936532484575635034309708812329356184178068005214278172082219184553984, 869292753515289797738996903425667808718790361916589577008016293615078398023922669/59404934841503635558701400709674780140155461004709195825540532695118232067810758305647985007001600, 1717531605744410360635635083389478316588313496630536848988455754354879369295068973/180388155799200064074593277764744149449984265733812070250726739525419997376644863635443271545651200, 31299907071900218897971026445896657679496819292586977817490723908705536712400282191/5050868362377601794088611777412836184599559440546737967020348706711759926546056181792411603278233600, 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2365807038708269681643405811853124264897142108207661676021322849554129541724538712828069/4993605946200703725748772887319195801798156435232196663439958010516810602578814036279976491158782083072000, 1743442550049309641060011406008600467181694819066702759814306064998734704691036735197/5643543729561152993580952965052742398952213556956407058780876539456405425169797283922973928160074137600, 9256655801374931608739126802876252957056848904251166546849608775226508061693870878057973/45941174705046474276888710563336601376543039204136209303647613696754657543725089133775783718660795164262400, 5860302328688239883089190420717746109476383007768745900617447034853291200216586721769111/44582913887679882863485044337985780118453940653753051811191945117894085059823651715907630113065606437666816, 28053455147941361649759906475266552733647037275384119733409357031222102043869587473860911/327065081069868027250332531303641779224870220196627487272531959501720395079871742556866589278993262628044800, 264759291857382731049897050412902587080661392902422783564415396842409102553457523174201863/4729328954586389559339914752974918362441454002352215793867135617344844010688489108821466966088225000521728000, 17738976089151435365104180865025447153088830155088668257513572834531253403779405903937081/485378497970708402142780724647425831934780805504569515686363918622233990570660724326413714940633618474598400, 57525628103894465056530448854456637470052052058655852991541497265019848592917947736293286079/2410583772321726208401906190888975651720895392457894042704757765445462890770129421294701073881162802792245493760, 55228426744890376132798337517678610636716441713442380162540491962510893335680047508448823064257/3543558145312937526350802100606794208029716226913104242775993915204830449432090249303210578605309320104600875827200, 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679218535878072604417972951902089523297666389232376275013785472046858414007981321736517084987758889/365627077498889340736753338913666639816749082113641497768835735594687013600098549870645543459720040173987901527713382400, 45417393373970549098858199347141034064760679726496049564545340505224419689363530500770466106523/37389347883545002756411639956822305859294830456521473901802265965330397127265880774180900013579934754372054354493440, 399259822519396731265204721252689123028743446447607436151433288023973123542819906261760233584027604061/502571037434836984721791862215876326729931465596205404205890574744697058693953643094923692464633364311881551918093303808000, 26101794801999407946521269456985493150728997931199220117672234351060842516480570759234027159081446027331/50228385398487422073052226686603868311465721904443614397491578013055723180327138387029802178322500238941761960271153620582400, 284144216407687599224893813056318301171542375833389731465664500352686421636271509935850233249178434713/835751177322955758109566906821472202362663110684553194152747073511864355219003287488936683560103803478141488142600057651200, 730938516408180437035841389422899083911122781231294232023591609987077693308745825319390674292796175194631/3285505028292003676280329424096571521928101220723115516853279295389841153236945723776507890411480072233269818186189346638397440, 1868937745740297460488979887265037261787387207295879321467067027349335616996817327148413832713240424820581/12835844017341519738872971585180153974557814804975612592724281118189701996517099781062342654295746518760803232340309633820262400, 1787876660846170745716890390888531262497790622062517509618700672314055212011820991775272488838221575117499909/18758669185343392418381499902341853594246635064985788117652770834154378774909990108609680764777926698131859580977338221997326336000, 2156882699076306255809731489112265890436793713082869472540030902073082421121441428170245959033338155775691587/34566369858526045634187570175777755516354866669155630120354631473844946185229396793019103448772606429522027417595395569546061414400, 1843656275251981391296493817574645252404230433198593603479662652164816956012478734748031859626149155631887483/45123126058562378490034044310542313282160542165411268508462937842884078398556212570346559366911267312051727682969124486731750440960, 2673074784867256927354914827764345412247861642958792920612642138276565398784825417674129359184316390038763034759/99897075812915241195895778098114900915281134981710498118327744840736016830109753870024382859643556694109620796907971924911434445619200, 45071101018309497968827687694094165484312402273370296244044733607293765163958268103216717107423586927884444592391/2571544080603431047558865513622441642915946635980805725755662592997011013884760760912240565225663169093531528901050374067074989922713600, 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777441138220252508990646307666406094259018498899013918743273680931312843069608681345297698749404761730925747575794661/240613695156200656588293705459569120182960786215107613135774099467314221913271634927402648945863224950339538405365291705892479588669535027200, 384053990211095873990547577275044200337234237227627041577626308376879703709870743493096979021990578998450287706073/181332266235666122175800969048779057073441403469189902471066438207749402532425832773743370269369238432865640043645617557970186913867366400, 772174896260605728589801483409364537345209555230876859738788954578971534688382498068015674541672558823400596506637/556117939498152519079261260692378552040741416521820369343083434824300975146236444362255737163012076156408794465405142617625145428357939200$

浮点结果为:
0.75000000000000000000000000000000000000
0.41666666666666666666666666666666666667
0.21527777777777777777777777777777777778
0.11134259259259259259259259259259259259
0.058294753086419753086419753086419753087
0.030946869488536155202821869488536155203
0.016651969429747207524985302763080540858
0.0090747315372765475646134082348074117621
0.0050044872784981238342006517589507987313
0.0027906145777730711429842659838087364379
0.0015722967511278053687747332008877504991
0.00089446520207703120940432325891859499541
0.00051345555031606954019151637110571941935
0.00029722324832383553477475474461028831128
0.00017339873992969325694407123954336156631
0.00010189372898755652484232797696845556953
6.0277483874553683157240377123355789585 E-5
3.5879783527775813447904402345231427088 E-5
2.1479590622591303087136679530235161455 E-5
1.2926827130702352979005508324490224054 E-5
7.8175339898533215358346056701232104773 E-6
4.7489253667381015466252788538935013069 E-6
2.8967950756408570688427682004200384627 E-6
1.7737766568213138273648289614743756272 E-6
1.0899654672068948499947329653737226157 E-6
6.7196143810592103711259713642930764763 E-7
4.1551754619080520574304384621287706311 E-7
2.5766354001711554593793363651163094677 E-7
1.6019504856086308041987672808044514298 E-7
9.9838858645948817097390362944375614436 E-8
6.2363947783373046758000691015024529970 E-8
3.9037915402043459608881930549469782859 E-8
2.4484996730285631061982824630239888339 E-8
1.5385815255741975562982274765836347109 E-8
9.6849932102839058568405667685850351547 E-9
6.1064948183192075649572441183141620701 E-9
3.8561846646779032462124966589661485744 E-9
2.4387062248456000897851448221478035297 E-9
1.5444103194093293730820054416033471056 E-9
9.7934539454781545825968430053218665196 E-10
6.2179890053643593920866500145814959093 E-10
3.9525556158191679153900061208888888303 E-10
2.5153307516045362412980939207744969231 E-10
1.6024267307220263145909273296972212991 E-10
1.0218937967449776326328252927298656003 E-10
6.5231574184559642879146981200952112840 E-11
4.1678859874210113619013780192528322096 E-11
2.6653992064456000824007883881645961120 E-11
1.7060072345252476949045218162607928396 E-11
1.0928394810162730166025581637383532680 E-11
7.0060697950606558197924053860327649285 E-12
4.4949167412884307229003533031739857014 E-12
2.8859265938445612714298111395077285866 E-12
1.8541864482702590722786390852018288659 E-12
1.1921056636132198386172829303237245329 E-12
7.6693521998039990733282939844074324576 E-13
4.9371364432537097234698495198993321167 E-13
3.1802009484131124815990844334336572142 E-13
2.0496889009123865885045617408485256302 E-13
1.3218029607656890387473034793418943885 E-13
8.5286980292628803526863005862575080161 E-14
5.5058993364617516622181307808728077804 E-14
3.5562800116152509758798100176772994212 E-14
2.2981521404941954772189167792385315355 E-14
1.4858332450532259456377144919439564907 E-14
9.6108868769511607694125704303349040351 E-15
6.2194628733965564337896056220505361517 E-15
4.0265445376803258411298857966309582563 E-15
2.6079352083207914159675632198879571232 E-15
1.6898199824657665164083155985938284214 E-15
1.0953637309104459296427159974540043819 E-15
7.1030631478567259219367903743130730231 E-16
4.6078436333983181564271983043636341785 E-16
2.9902675941357638716827624343031815929 E-16
1.9412360445492463836767188790543926381 E-16
1.2606614931971982113573856154300512337 E-16
8.1896721628108748550520467434398101931 E-17
5.3220460585889027074051291741523343705 E-17
3.4596420094398854027634040190461049188 E-17
2.2496792079755385385993257024599771903 E-17
1.4633342429120095365463042003011944851 E-17
9.5213102996423382626613893960792869976 E-18
6.1969358190056596875653257332472823557 E-18
4.0344221462775593345250483483164412589 E-18
2.6272827577978227946453780508728735212 E-18
1.7113963616273097302879942016563260661 E-18
1.1150901503865710409267674983475042576 E-18
7.2674543019297147597148534154224268833 E-19
4.7376726641962045319448129514392040033 E-19
3.0892691429271184057957880168981421207 E-19
2.0148931455943204181383318107545406105 E-19
1.3144727021325731805501510288939121670 E-19
8.5773311709630504914342035673687734186 E-20
5.5982422538111994263910246509762637930 E-20
3.6546687097420508781119197369070585813 E-20
2.3863774727268371943295884159771663174 E-20
1.5585585019379740334443848545601089258 E-20
1.0181177810291582861438418535379085533 E-20
6.6521501460518411296694813101916019605 E-21
4.3472377036631031195895537955761539242 E-21
2.8415174394843821150179077919061523246 E-21
1.8576811666256743668385573036391467471 E-21
1.2147147769313911303869059999416615182 E-21
7.9443460283197175966806823991059893699 E-22
5.1966223072711864566897963724453053129 E-22
3.3998661816767858791474166417191630179 E-22
2.2247371716492691781932421445036942677 E-22
1.4560302721155848342034073020971191567 E-22
9.5309355007074945429111473967440003090 E-23
6.2398299500469399356757328851251835429 E-23
4.0858345515761020119171524214972026641 E-23
2.6758288599691595538180003261954575312 E-23
1.7526863085206474320634914157610200131 E-23
1.1481989896431875055970625367399567366 E-23
7.5230866756719958460393059273702887710 E-24
4.9299188755088812228281322147060078634 E-24
3.2310760105136838310958331670440744493 E-24
2.1179572625644301990243346871789759075 E-24
1.3885092377300858011878203533297969735 E-24

n次只有一人的概率极限情况每增加一人，概率减半。而n次余两人的概率好像是每增加一人，概率是原先的2/3.

mathe 发表于 2021-10-27 16:06:26

上面Fans提到n趋于无穷时，n次只有k人的概率好像是每增加1人，概率是原先的k/(k+1)， 我也觉得是这样。
为此我计算了$k<=3,n<1000$的情况，结果如下，和猜测的匹配程度还是挺好的。

比如k=1, n=11/n=10 为0.4545
   k=1,n=101/n=100为0.4950
    k=1,n=999/n=998为0.4995

k=2,n=11/n=10为0.5631
k=2,n=101/n=100为0.6537
k=2,n=999/n=998为0.6653

k=3,n=11/n=10为0.6181
k=3,n=101/n=100为0.7286
k=3,n=999/n=998为0.7478
我们可以猜测n次k个人的概率极限为$\frac{c_k}{n^k}(\frac{k}{k+1})^n$, 其中$c_1=2,c_2=30, c_3=600$

mathe 发表于 2021-10-27 19:26:39

根据14#,我们有
$f_{n,h}(x_1,x_2,...,x_h) = f_{n-1,h} (x_1,x_2,...,x_h) \frac{\sum_{t=1}^h x_t}{h+1} +\sum_{t=0}^{h-1} \frac{(t+1)f_{n-1,h-1}(x_1,x_2,...,x_t, x_{t+2},...,x_h)}{h}$
由于在n充分大时$f_{n-1,h-1}$要比$f_{n,h}$以远远更加的速度指数衰减，所以对于充分大的n,我们可以认为
$f_{n,h}(x_1,x_2,...,x_h) = f_{n-1,h} (x_1,x_2,...,x_h) \frac{\sum_{t=1}^h x_t}{h+1}$
如果对于任意的初始正函数$g(x_1,x_2,...,x_h)$我们能够证明
$\lim_{n\to\infty}\frac{\int_{0\lt x_1\lt x_2\cdots\lt x_h} g(x_1,x_2,...,x_h) (x_1+x_2+...+x_h)^n dx_1dx_2\cdots\dx_h}{\int_{0\lt x_1\lt x_2\cdots\lt x_h}g(x_1,x_2,...,x_h)(x_1+x_2+...+x_h)^{n+1}dx_1dx_2\cdots\dx_h}=\frac{1}{h}$,和g的初始选择无关，那么就可以证明上述每增加一个人概率会减少$k/{k+1}$的结论。
这个极限的几何意义是当n越来越大时，$x_1+x_2+...+x_h$绝对值越大的部分起的作用越大，也就是最终极限应该收敛到
$\lim_{x_i\to 1} \frac{g(x_1,x_1,...,x_1) (x_1+x_1+...+x_1)^n }{g(x_1,x_1,...,x_1)(x_1+x_1+...+x_1)^{n+1}}=1/h$

mathe 发表于 2021-10-28 08:42:07

如果我们选择上面g=1分别进行计算，可以得到h=1时，结果为
\(\int_0^1 (\frac{x_1}2)^n dx_1=\frac1{(n+1)2^n}\)
而h=2结果为
\(\int_0^1 dx_2 \int_0^{x_2}(\frac{x_1+x_2}3)^ndx_1=\frac{2^{n+1}-1}{(n+1)(n+2)3^n}\)
h=3的结果为
\(\int_0^1 dx_3\int_0^{x_3}dx_2 \int_0^{x_2}(\frac{x_1+x_2+x_3}4)^ndx_1=\frac{3^{n+2}-2^{n+3}+1}{2(n+1)(n+2)(n+3)4^n}\)
这个结果和前面的预测基本一致（除了乘数$c_k$会不同,这时因为g不是1，g(1,1,...,1)要远远大于1)
而一般的结果是近似于\(\frac{h^{n+h-1}}{(h-1)!(n+1)(n+2)...(n+h)(h+1)^n}\)

mathe 发表于 2021-10-28 09:55:26

现在可以算出前面c2=30,c3=600估计的不准。应该是c2=27,c3=576

mathe 发表于 2021-10-28 12:53:36

现在假设$u_{m,h}=f_{m,h}(1,1,...,1)$
根据17#的结论，我们知道对于充分大的n,n次后正好h个人的概率接近
\(\int_0^1 dx_h\int_0^{x_{h-1}} dx_{h-2}...\int_0^{x_2}f_{m,h}(x_1,x_2,...,x_h)(\frac{x_1+x_2+...+x_h}{h+1})^{n-m}dx_1\approx \frac{u_{m,h}h^{n-m+h-1}}{(h-1)!(n-m+1)(n-m+2)...(n-m+h)(h+1)^{n-m}}\)
即
\(\frac{(h+1)^m u_{m,h}}{(h-1)!h^{m+1-h}}(\frac{h}{h+1})^n\frac1{(n-m+1)(n-m+2)...(n-m+h)}\approx \frac{(h+1)^m u_{m,h}}{(h-1)!h^{m+1-h}}(\frac{h}{h+1})^n\frac1{n^h}\)
也就是我们前面求的\(c_h \approx \frac{(h+1)^m u_{m,h}}{(h-1)!h^{m+1-h}}\)
另外16#第一行公式中将$x_i=1$代入可以得出递推式\(u_{m,h}=\frac{h u_{m-1,h}}{h+1}+\frac{(h+1)u_{m-1,h-1}}2\)而且我们已经知道$u_{1,1}=1$,由此可以轻松计算$c_h$
设\(\lambda_{m,h}=\frac{(h+1)^m u_{m,h}}{(h-1)!h^{m+1-h}}, \lambda_{m,1}=2\)
递推式变化为\(\lambda_{m,h}=\lambda_{m-1,h}+\frac{(h+1)}2 (\frac h{h-1})^h (\frac{h^2-1}{h^2})^m \lambda_{m-1,h-1}\)
特别的\(\lambda_{h,h}=\frac{(h+1)}2 (\frac h{h-1})^h (\frac{h^2-1}{h^2})^h \lambda_{h-1,h-1}\)
而且\(c_h = \lim_{m\to\infty}\lambda_{m,h}\)
比如让h=2代入，我们可以化简得到\(\lambda_{2,2}=\frac{27}4, \lambda_{m,2}=\lambda_{m-1,2}+12(\frac34)^m\),可以得到\(c_2=\lim_{m\to\infty} \lambda_{m,2}=27\)

mathe 发表于 2021-10-28 14:31:41

计算结果$c_3$好像是864而不是576，和前面的模拟结果有偏差，看来还有问题

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