Statistical Analysis of the Number of Donors a Beggar Needs Before He (or She) can go home By Shalosh B. Ekhad In a certain country there are only, 4, types of coins, of denominations, 1, 5, 10, 25 A person passing a beggar throws A coin of, 1, cents , with probability, 1/4 A coin of, 5, cents , with probability, 1/4 A coin of, 10, cents , with probability, 1/4 A coin of, 25, cents , with probability, 1/4 Each person's donation is independent of the others. The beggar decides to go home as soon as the amount in his hat exceeds n cen\ ts. Let X(n) be the random variable "Number of donors until he goes home". Since the expected size of a single donation is 41/4 4 n The expectation of X(n) is roughly, --- 41 More precisely, up to exponentially small error, the Expectation of X(n) is 1420 4 n ---- + --- 1681 41 and in Maple notation it is 1420/1681+4/41*n The variance of X(n), up to exponentially small error is 1318660 5292 n ------- + ------ 2825761 68921 and in Maple notation it is 1318660/2825761+5292/68921*n The , 3, -th moment about the mean of of X(n), up to exponentially small error is 407119260 15039924 n ---------- + ---------- 4750104241 115856201 and in Maple notation it is 407119260/4750104241+15039924/115856201*n The limit of the scaled , 3, -th moment as n goes to infinity is 0 The , 4, -th moment about the mean of of X(n), up to exponentially small error is 6554942355308 90446206956 84015792 2 - ------------- + ------------ n + ---------- n 7984925229121 194754273881 4750104241 and in Maple notation it is -6554942355308/7984925229121+90446206956/194754273881*n+84015792/4750104241*n^2 The limit of the scaled , 4, -th moment as n goes to infinity is 3 The , 5, -th moment about the mean of of X(n), up to exponentially small error is 795912778080 2 98702717596498500 360414703824180 ------------- n - ----------------- + --------------- n 7984925229121 13422659310152401 327381934393961 and in Maple notation it is 795912778080/7984925229121*n^2-98702717596498500/13422659310152401+ 360414703824180/327381934393961*n The limit of the scaled , 5, -th moment as n goes to infinity is 0 The , 6, -th moment about the mean of of X(n), up to exponentially small error is 7779789083244240 2 780179095560685120460 2223057856320 3 ----------------- n - --------------------- + --------------- n 13422659310152401 22563490300366186081 327381934393961 585618363452440812 + ------------------ n 550329031716248441 and in Maple notation it is 7779789083244240/13422659310152401*n^2-780179095560685120460/ 22563490300366186081+2223057856320/327381934393961*n^3+585618363452440812/ 550329031716248441*n The limit of the scaled , 6, -th moment as n goes to infinity is 15 The , 7, -th moment about the mean of of X(n), up to exponentially small error is 3642168393880832514525540 64426788749956852800 2 44225689426793280 3 - ------------------------- + -------------------- n + ------------------ n 37929227194915558802161 22563490300366186081 550329031716248441 15464234474027876251596 - ----------------------- n 925103102315013629321 and in Maple notation it is -3642168393880832514525540/37929227194915558802161+64426788749956852800/ 22563490300366186081*n^2+44225689426793280/550329031716248441*n^3-\ 15464234474027876251596/925103102315013629321*n The limit of the scaled , 7, -th moment as n goes to infinity is 0 The , 8, -th moment about the mean of of X(n), up to exponentially small error is 409802245917434125952112 2 702935156681319139200 3 ------------------------ n + --------------------- n 37929227194915558802161 925103102315013629321 21457499297956619538596787412 82350955229518080 4 + ----------------------------- + -------------------- n 63759030914653054346432641 22563490300366186081 290559089322037413334040724 - --------------------------- n 1555098314991537910888601 and in Maple notation it is 409802245917434125952112/37929227194915558802161*n^2+702935156681319139200/ 925103102315013629321*n^3+21457499297956619538596787412/ 63759030914653054346432641+82350955229518080/22563490300366186081*n^4-\ 290559089322037413334040724/1555098314991537910888601*n The limit of the scaled , 8, -th moment as n goes to infinity is 105 The , 9, -th moment about the mean of of X(n), up to exponentially small error is 278808845199530369802896160 2 906212414001940395888954263761020 --------------------------- n + --------------------------------- 63759030914653054346432641 107178930967531784356353269521 9487319912274918165298560 3 2808508181359080453120 4 + ------------------------- n + ----------------------- n 1555098314991537910888601 37929227194915558802161 3249973458285346354947101546700 - ------------------------------- n 2614120267500775228203738281 and in Maple notation it is 278808845199530369802896160/63759030914653054346432641*n^2+ 906212414001940395888954263761020/107178930967531784356353269521+ 9487319912274918165298560/1555098314991537910888601*n^3+2808508181359080453120/ 37929227194915558802161*n^4-3249973458285346354947101546700/ 2614120267500775228203738281*n The limit of the scaled , 9, -th moment as n goes to infinity is 0 The , 10, -th moment about the mean of of X(n), up to exponentially small error is 52807287354097858827196395694320 2 3922211295671487114240 5 - -------------------------------- n + ------------------------- n 107178930967531784356353269521 1555098314991537910888601 107272489420510662598480347840 3 67473119234550258497318400 4 + ------------------------------ n + -------------------------- n 2614120267500775228203738281 63759030914653054346432641 15508165707585540235172099657006472820 + -------------------------------------- 180167782956420929503029846064801 18305804874482113565021003461151508 - ----------------------------------- n 4394336169668803158610484050361 and in Maple notation it is -52807287354097858827196395694320/107178930967531784356353269521*n^2+ 3922211295671487114240/1555098314991537910888601*n^5+ 107272489420510662598480347840/2614120267500775228203738281*n^3+ 67473119234550258497318400/63759030914653054346432641*n^4+ 15508165707585540235172099657006472820/180167782956420929503029846064801-\ 18305804874482113565021003461151508/4394336169668803158610484050361*n The limit of the scaled , 10, -th moment as n goes to infinity is 945 ----------------------- This ends this article that took, 0.944, seconds to produce.