Statistical Analysis of the Number of Donors a Beggar Needs Before He (or Sh\ e) can go home By Shalosh B. Ekhad In a certain country there are only, 2, types of coins, of denominations, 1, 2 A person passing a beggar throws A coin of, 1, cents , with probability, 1/2 A coin of, 2, cents , with probability, 1/2 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 3/2 2 n The expectation of X(n) is roughly, --- 3 More precisely, up to exponentially small error, the Expectation of X(n) is 2 n 2/9 + --- 3 and in Maple notation it is 2/9+2/3*n The variance of X(n), up to exponentially small error is 2 n 2/81 + --- 27 and in Maple notation it is 2/81+2/27*n The , 3, -th moment about the mean of of X(n), up to exponentially small error is 26 2 n - --- + --- 729 81 and in Maple notation it is -26/729+2/81*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 62 2 - ---- + 2/243 n + 4/243 n 2187 and in Maple notation it is -62/2187+2/243*n+4/243*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 370 40 2 730 - ----- + ---- n - ----- n 59049 2187 19683 and in Maple notation it is -370/59049+40/2187*n^2-730/19683*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 21862 40 3 20 2 3986 ------ + ---- n + ---- n - ----- n 531441 6561 2187 59049 and in Maple notation it is 21862/531441+40/6561*n^3+20/2187*n^2-3986/59049*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 148778 280 3 7000 2 6706 ------- + ----- n - ------ n - ------ n 1594323 19683 177147 177147 and in Maple notation it is 148778/1594323+280/19683*n^3-7000/177147*n^2-6706/177147*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 2788214 560 4 7280 3 200956 2 2142338 -------- + ------ n + ------ n - ------- n + -------- n 43046721 177147 531441 1594323 14348907 and in Maple notation it is 2788214/43046721+560/177147*n^4+7280/531441*n^3-200956/1594323*n^2+2142338/ 14348907*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 99560690 2240 4 67760 3 73640 2 7782470 - --------- + ------ n - ------- n - ------ n + -------- n 387420489 177147 1594323 531441 14348907 and in Maple notation it is -99560690/387420489+2240/177147*n^4-67760/1594323*n^3-73640/531441*n^2+7782470/ 14348907*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 419748842 1120 5 11200 4 1083880 3 15247420 2 3187294 - --------- + ------ n + ------ n - ------- n + -------- n + ------- n 387420489 531441 531441 4782969 43046721 4782969 and in Maple notation it is -419748842/387420489+1120/531441*n^5+11200/531441*n^4-1083880/4782969*n^3+ 15247420/43046721*n^2+3187294/4782969*n The limit of the scaled , 10, -th moment as n goes to infinity is 945 ----------------------- This ends this article that took, 0.088, seconds to produce.