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, 6, types of coins, of denominations, 1, 2, 3, 4, 5, 6 A person passing a beggar throws A coin of, 1, cents , with probability, 1/6 A coin of, 2, cents , with probability, 1/6 A coin of, 3, cents , with probability, 1/6 A coin of, 4, cents , with probability, 1/6 A coin of, 5, cents , with probability, 1/6 A coin of, 6, cents , with probability, 1/6 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 7/2 2 n The expectation of X(n) is roughly, --- 7 More precisely, up to exponentially small error, the Expectation of X(n) is 10 2 n -- + --- 21 7 and in Maple notation it is 10/21+2/7*n The variance of X(n), up to exponentially small error is 50 10 n --- + ---- 441 147 and in Maple notation it is 50/441+10/147*n The , 3, -th moment about the mean of of X(n), up to exponentially small error is 82 50 n - ---- + ---- 9261 1029 and in Maple notation it is -82/9261+50/1029*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 1790 86 100 2 - ----- + ---- n + ---- n 64827 1029 7203 and in Maple notation it is -1790/64827+86/1029*n+100/7203*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 3595190 5000 2 98150 - -------- + ------ n + ------- n 28588707 151263 1361367 and in Maple notation it is -3595190/28588707+5000/151263*n^2+98150/1361367*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 149801510 5000 3 4300 2 1161770 - --------- + ------- n + ----- n + -------- n 600362847 1058841 50421 66706983 and in Maple notation it is -149801510/600362847+5000/1058841*n^3+4300/50421*n^2+1161770/66706983*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 217614014 25000 3 1627000 2 2018290 - --------- + ------- n + ------- n - ------- n 600362847 1058841 9529569 7411887 and in Maple notation it is -217614014/600362847+25000/1058841*n^3+1627000/9529569*n^2-2018290/7411887*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 738969110 50000 4 6418000 3 345052580 2 15059975942 ----------- + -------- n + -------- n + ---------- n - ----------- n 37822859361 22235661 66706983 1400846643 12607619787 and in Maple notation it is 738969110/37822859361+50000/22235661*n^4+6418000/66706983*n^3+345052580/ 1400846643*n^2-15059975942/12607619787*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 326332848650 1000000 4 149710000 3 49034200 2 14596263190 ------------ + -------- n + --------- n - --------- n - ----------- n 113468578083 51883209 466948881 363182463 4202539929 and in Maple notation it is 326332848650/113468578083+1000000/51883209*n^4+149710000/466948881*n^3-49034200 /363182463*n^2-14596263190/4202539929*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 188566321892170 500000 5 14200000 4 19695859000 3 --------------- + --------- n + --------- n + ----------- n 12973240760823 363182463 121060821 22880495169 636238458500 2 1315331665850 - ------------ n - ------------- n 205924456521 205924456521 and in Maple notation it is 188566321892170/12973240760823+500000/363182463*n^5+14200000/121060821*n^4+ 19695859000/22880495169*n^3-636238458500/205924456521*n^2-1315331665850/ 205924456521*n The limit of the scaled , 10, -th moment as n goes to infinity is 945 ----------------------- This ends this article that took, 0.154, seconds to produce.