Studies in the Classical Gambler's Ruin Problem by Shalosh B. Ekhad Consider A Player that starts at x=, n, and with prob. 1/2 walks, at each day, one step to the left or one step to the right, until it either reaches x=0 or x=, A Let D be the random variable: duration of a WINNING game The Expected value of D is (n - A) (n + A) - --------------- 3 The , 2, -th moment (about the mean) of D is 2 2 2 (n - A) (n + A) (n - 5 + A ) - ------------------------------- 45 and setting n=, A x, it is 2 2 2 2 2 A (x - 1) (x + 1) (A x - 5 + A ) - ------------------------------------- 45 and asymptotically it is, as, A, goes to infinity 2 4 2 (x - 1) (x + 1) (x + 1) A - ----------------------------- 45 If it starts at the middle, i.e., x = 1/2, it is 4 A ---- 24 which is roughly 4 0.04166666667 A The , 3, -th moment (about the mean) of D is 4 2 2 2 2 4 4 (n - A) (n + A) (4 n - 21 n + 4 n A + 21 - 21 A + 4 A ) - -------------------------------------------------------------- 945 and setting n=, A x, it is 2 4 4 2 2 4 2 2 4 4 A (x - 1) (x + 1) (4 A x - 21 A x + 4 A x + 21 - 21 A + 4 A ) - ----------------------------------------------------------------------- 945 and asymptotically it is, as, A, goes to infinity 4 2 6 4 (x - 1) (x + 1) (4 x + 4 x + 4) A - -------------------------------------- 945 If it starts at the middle, i.e., x = 1/2, it is 6 A ---- 60 which is roughly 6 0.01666666667 A The , 4, -th moment (about the mean) of D is 6 4 4 2 2 2 2 2 4 - 4 (n + A) (n - A) (5 n - 10 n + 5 n A - 49 n - 80 n A + 19 n A + 30 2 4 6 + 301 A - 150 A + 19 A )/4725 and setting n=, A x, it is 2 6 6 4 4 6 4 2 2 4 2 - 4 A (x + 1) (x - 1) (5 A x - 10 A x + 5 A x - 49 A x - 80 A x 6 2 2 4 6 + 19 A x + 30 + 301 A - 150 A + 19 A )/4725 and asymptotically it is, as, A, goes to infinity 6 4 2 8 4 (x + 1) (x - 1) (5 x + 5 x + 19 x + 19) A - ----------------------------------------------- 4725 If it starts at the middle, i.e., x = 1/2, it is 8 103 A ------ 6720 which is roughly 8 0.01532738095 A The , 5, -th moment (about the mean) of D is 8 6 6 2 4 4 4 2 4 - 16 (n - A) (n + A) (4 n + 55 n + 4 n A + 48 n A - 165 n A - 462 n 2 2 4 2 2 2 6 2 6 4 + 792 n - 627 n A + 924 n A + 92 n A - 1518 A - 847 A + 2310 A 8 - 297 + 92 A )/93555 and setting n=, A x, it is 2 8 8 6 6 8 6 8 4 6 4 - 16 A (x - 1) (x + 1) (4 A x + 55 A x + 4 A x + 48 A x - 165 A x 4 4 2 2 6 2 4 2 8 2 2 - 462 A x + 792 A x - 627 A x + 924 A x + 92 A x - 1518 A 6 4 8 - 847 A + 2310 A - 297 + 92 A )/93555 and asymptotically it is, as, A, goes to infinity 8 6 4 2 10 16 (x - 1) (x + 1) (4 x + 4 x + 48 x + 92 x + 92) A - --------------------------------------------------------- 93555 If it starts at the middle, i.e., x = 1/2, it is 10 229 A ------- 15120 which is roughly 10 0.01514550265 A The , 6, -th moment (about the mean) of D is 10 8 8 2 6 6 4 8 (n - A) (n + A) (175 n - 116025 n + 175 n A + 795795 n - 44870 n A 6 2 4 6 4 4 4 2 4 + 109200 n A - 136390 n A + 679770 n A - 135135 n A - 1525095 n 2 2 6 2 4 2 8 2 2 + 594594 n + 2511600 n A - 4927923 n A - 307561 n A - 1480050 n A 6 4 8 2 10 - 12165153 A + 14330745 A + 737100 + 3367455 A + 774774 A - 307561 A )/127702575 and setting n=, A x, it is 2 10 10 8 8 10 8 6 6 8 A (x - 1) (x + 1) (175 A x - 116025 A x + 175 A x + 795795 A x 10 6 8 6 10 4 8 4 - 44870 A x + 109200 A x - 136390 A x + 679770 A x 6 4 4 4 2 2 8 2 - 135135 A x - 1525095 A x + 594594 A x + 2511600 A x 6 2 10 2 4 2 6 - 4927923 A x - 307561 A x - 1480050 A x - 12165153 A 4 8 2 10 + 14330745 A + 737100 + 3367455 A + 774774 A - 307561 A )/127702575 and asymptotically it is, as, A, goes to infinity 8 (x - 1) (x + 1) 10 8 6 4 2 12 (175 x + 175 x - 44870 x - 136390 x - 307561 x - 307561) A / 127702575 If it starts at the middle, i.e., x = 1/2, it is 12 98443 A --------- 5322240 which is roughly 12 0.01849653529 A The , 7, -th moment (about the mean) of D is 2 8 2 4 16 (n + A) (n - A) (-131220 + 265356 n - 1732367 A - 455364 A - 132561 n 6 2 6 4 4 6 2 4 - 10725 n - 904540 n A - 61698 n A + 56420 n A - 1261689 A 10 8 2 2 12 12 10 - 3815 n + 21385 n + 879450 n A + 76 n - 31124 A + 391281 A 6 2 4 4 2 10 2 8 4 + 3006003 A + 470613 n A - 418275 n A + 76 n A - 652 n A 6 6 4 8 2 10 2 8 4 6 - 3512 n A - 14380 n A - 31124 n A + 307561 n A + 96350 n A 6 4 8 2 + 4830 n A - 175 n A )/18243225 and setting n=, A x, it is 2 4 2 8 2 16 A (x + 1) (x - 1) (-131220 + 879450 A x - 1732367 A - 455364 A 10 2 2 2 8 6 8 4 8 2 + 307561 A x + 265356 A x + 56420 A x - 61698 A x - 904540 A x 4 4 4 12 10 12 6 12 4 - 1261689 A - 132561 A x + 76 A x - 3512 A x - 14380 A x 12 2 8 8 12 12 12 12 8 - 31124 A x + 21385 A x + 76 A x - 31124 A - 652 A x 10 6 10 8 10 6 10 10 + 391281 A + 3006003 A - 175 A x + 4830 A x - 3815 A x 6 4 6 2 6 6 10 4 - 418275 A x + 470613 A x - 10725 A x + 96350 A x )/18243225 and asymptotically it is, as, A, goes to infinity 16 (x + 1) (x - 1) 10 6 4 2 12 8 14 (76 x - 3512 x - 14380 x - 31124 x + 76 x - 31124 - 652 x ) A / 18243225 If it starts at the middle, i.e., x = 1/2, it is 14 907061 A ---------- 34594560 which is roughly 14 0.02621975825 A The , 8, -th moment (about the mean) of D is 2 8 2 16 (n + A) (n - A) (-156151800 - 77216652 n + 6607829280 A - 176746212 A 4 6 12 2 2 6 + 310403340 n - 177766875 n + 31045 n A + 1867671000 n A 4 4 6 2 2 12 8 6 - 388491480 n A - 288139800 n A - 38606453 n A - 522815 n A 10 4 4 4 10 6 8 + 42945 n A - 287728740 A - 17692305 n A - 4679825 n A 10 8 14 2 2 12 + 2845290 n + 29835000 n + 31045 n - 585959400 n A - 1025780 n 12 10 6 2 4 + 549021460 A - 2887285746 A - 5410479789 A + 938368431 n A 4 2 10 2 8 4 6 6 + 738963225 n A - 1085280 n A - 6004740 n A + 19034560 n A 4 8 2 10 2 8 4 6 + 147148260 n A + 444450720 n A - 1674035526 n A - 258595500 n A 6 4 8 2 14 + 68422620 n A + 39323550 n A - 38606453 A )/13956067125 and setting n=, A x, it is 2 4 2 8 16 A (x + 1) (x - 1) (-156151800 - 585959400 A x + 6607829280 A 2 10 2 14 12 14 2 - 176746212 A - 1674035526 A x + 31045 A x - 38606453 A x 14 8 2 2 8 6 8 4 - 522815 A x - 77216652 A x - 288139800 A x - 388491480 A x 8 2 4 4 4 12 10 + 1867671000 A x - 287728740 A + 310403340 A x - 1085280 A x 12 6 12 4 12 2 8 8 + 19034560 A x + 147148260 A x + 444450720 A x + 29835000 A x 12 12 12 12 8 10 - 1025780 A x + 549021460 A - 6004740 A x - 2887285746 A 6 10 8 10 6 10 10 - 5410479789 A + 39323550 A x + 68422620 A x + 2845290 A x 6 4 6 2 6 6 10 4 + 738963225 A x + 938368431 A x - 177766875 A x - 258595500 A x 14 14 14 14 10 14 4 - 38606453 A + 31045 A x + 42945 A x - 17692305 A x 14 6 - 4679825 A x )/13956067125 and asymptotically it is, as, A, goes to infinity 12 2 8 14 16 (x + 1) (x - 1) (31045 x - 38606453 x - 522815 x - 38606453 + 31045 x 10 4 6 16 + 42945 x - 17692305 x - 4679825 x ) A /13956067125 If it starts at the middle, i.e., x = 1/2, it is 16 70588909 A ------------ 1660538880 which is roughly 16 0.04250963940 A The , 9, -th moment (about the mean) of D is 16 2 8 128 (n + A) (n - A) (597940 n - 46619640396 n - 1186392098619 A 2 4 6 12 2 + 77626876824 A + 16886358927 n + 9237632040 n - 37160865 n A 2 6 4 4 6 2 - 45950076900 n A + 136274119254 n A + 12015270540 n A 2 12 8 6 10 4 + 46211924241 n A - 247162545 n A - 273616245 n A 4 4 10 6 8 10 + 133410919023 A + 16066834245 n A + 2284456545 n A + 1574096895 n 8 14 2 2 12 - 7321061475 n - 13963005 n - 105425639910 n A - 60395832 n 12 10 6 14 2 - 339017270256 A + 962886891987 A + 331711938558 A + 597940 n A 4 12 10 6 12 4 6 10 - 1611566120 n A + 5915812 n A + 5237512 n A - 419459684 n A 8 8 2 4 4 2 - 54398624 n A - 257520420612 n A + 30165154110 n A 2 14 10 2 8 4 - 3511592948 n A + 1107636768 n A + 3818611944 n A 6 6 4 8 2 10 + 4284197064 n A - 42623503776 n A - 213326055432 n A 2 8 4 6 16 + 367773875307 n A - 18402792570 n A - 3511592948 A 6 4 8 2 14 - 31372663770 n A - 7409579625 n A + 55712058381 A + 23720530050)/ 5568470782875 and setting n=, A x, it is 2 16 2 16 8 16 6 128 A (x + 1) (x - 1) (-3511592948 A x - 54398624 A x - 419459684 A x 16 12 16 10 16 4 16 14 + 5237512 A x + 5915812 A x - 1611566120 A x + 597940 A x 4 2 8 16 16 2 - 105425639910 A x - 1186392098619 A + 597940 A x + 77626876824 A 10 2 14 12 14 2 + 367773875307 A x - 37160865 A x + 46211924241 A x 14 8 2 2 8 6 - 247162545 A x - 46619640396 A x + 12015270540 A x 8 4 8 2 4 + 136274119254 A x - 45950076900 A x + 133410919023 A 4 4 12 10 12 6 + 16886358927 A x + 1107636768 A x + 4284197064 A x 12 4 12 2 8 8 - 42623503776 A x - 213326055432 A x - 7321061475 A x 12 12 12 12 8 - 60395832 A x - 339017270256 A + 3818611944 A x 10 6 10 8 + 962886891987 A + 331711938558 A - 7409579625 A x 10 6 10 10 6 4 - 31372663770 A x + 1574096895 A x + 30165154110 A x 6 2 6 6 16 - 257520420612 A x + 9237632040 A x - 3511592948 A 10 4 14 14 14 - 18402792570 A x + 55712058381 A - 13963005 A x 14 10 14 4 14 6 - 273616245 A x + 16066834245 A x + 2284456545 A x + 23720530050 )/5568470782875 and asymptotically it is, as, A, goes to infinity 2 8 6 12 128 (x + 1) (x - 1) (-3511592948 x - 54398624 x - 419459684 x + 5237512 x 10 4 14 16 18 + 5915812 x - 1611566120 x + 597940 x + 597940 x - 3511592948) A / 5568470782875 If it starts at the middle, i.e., x = 1/2, it is 18 105218051 A ------------- 1357171200 which is roughly 18 0.07752747111 A The , 10, -th moment (about the mean) of D is 16 2 8 32 (n + A) (n - A) (-35999425 n + 278642266056 n + 42615729542295 A 2 4 6 18 + 896061856536 A - 1285174176660 n + 659386205280 n - 104367982199 A 12 2 2 6 4 4 + 6867717010 n A - 11227981974300 n A - 4412135907270 n A 6 2 2 12 8 6 + 1475508024900 n A - 8428606073734 n A + 138706898330 n A 10 4 4 18 + 34176572430 n A + 1145824806060 A + 2848505 n 4 10 6 8 10 - 1976025981930 n A + 33006143170 n A - 54210448215 n 8 14 2 2 12 - 33496997625 n - 1089510730 n + 2394725553000 n A + 14677123230 n 2 16 4 14 6 12 - 104367982199 n A - 47886741460 n A - 12499998420 n A 8 10 10 8 12 - 1572356090 n A + 158303750 n A + 42837832108830 A 12 6 14 4 16 2 10 + 134676300 n A + 48267340 n A + 2848505 n A - 70414679777079 A 6 14 2 4 12 + 3764555255736 A - 263093600 n A + 556102100180 n A 10 6 12 4 6 10 - 6741491680 n A - 2217456780 n A + 89375969760 n A 8 8 2 4 4 2 - 6473133590 n A + 3844225311696 n A - 3574467083880 n A 2 14 10 2 8 4 + 1545100897120 n A - 28870754220 n A - 332480750910 n A 6 6 4 8 2 10 - 1281242071560 n A + 954706910850 n A + 19447330790580 n A 2 8 4 6 16 - 12584216226219 n A + 5803351281090 n A + 1827507100815 A 6 4 8 2 14 + 1762294146690 n A - 163952740875 n A - 12630478454594 A + 765990918000)/20417726203875 and setting n=, A x, it is 2 16 2 16 8 32 A (x + 1) (x - 1) (1545100897120 A x - 6473133590 A x 16 6 16 12 16 10 + 89375969760 A x - 2217456780 A x - 6741491680 A x 16 4 16 14 4 2 + 556102100180 A x - 263093600 A x + 2394725553000 A x 8 16 16 2 + 42615729542295 A - 35999425 A x + 896061856536 A 18 10 2 14 12 - 104367982199 A - 12584216226219 A x + 6867717010 A x 14 2 14 8 2 2 - 8428606073734 A x + 138706898330 A x + 278642266056 A x 8 6 8 4 8 2 + 1475508024900 A x - 4412135907270 A x - 11227981974300 A x 4 4 4 12 10 + 1145824806060 A - 1285174176660 A x - 28870754220 A x 12 6 12 4 12 2 - 1281242071560 A x + 954706910850 A x + 19447330790580 A x 8 8 12 12 12 - 33496997625 A x + 14677123230 A x + 42837832108830 A 12 8 18 18 18 2 - 332480750910 A x + 2848505 A x - 104367982199 A x 18 4 18 6 18 8 - 47886741460 A x - 12499998420 A x - 1572356090 A x 10 18 10 18 14 - 70414679777079 A + 158303750 A x + 48267340 A x 18 16 18 12 6 + 2848505 A x + 134676300 A x + 3764555255736 A 10 8 10 6 10 10 - 163952740875 A x + 1762294146690 A x - 54210448215 A x 6 4 6 2 6 6 - 3574467083880 A x + 3844225311696 A x + 659386205280 A x 16 10 4 14 + 1827507100815 A + 5803351281090 A x - 12630478454594 A 14 14 14 10 + 765990918000 - 1089510730 A x + 34176572430 A x 14 4 14 6 - 1976025981930 A x + 33006143170 A x )/20417726203875 and asymptotically it is, as, A, goes to infinity 18 2 32 (x + 1) (x - 1) (-104367982199 + 2848505 x - 104367982199 x 4 6 8 10 - 47886741460 x - 12499998420 x - 1572356090 x + 158303750 x 14 16 12 20 + 48267340 x + 2848505 x + 134676300 x ) A /20417726203875 If it starts at the middle, i.e., x = 1/2, it is 20 674105568709 A ---------------- 4290832465920 which is roughly 20 0.1571036795 A The asymptotic Skewness is / 4 2 2 \1/2 | 10 (x + x + 1) | 4 |- -------------------------| | 2 3| \ (x - 1) (x + 1) (x + 1) / ---------------------------------- 7 and when, x = 1/2, it is 1/2 1/2 4 294 25 -------------- 175 which is roughly, 1.959591795 The asymptotic Kurtosis is 4 3 (5 x + 19) - -------------------------- 2 7 (x - 1) (x + 1) (x + 1) and when, x = 1/2, it is 309 --- 35 which is roughly, 8.828571429 The sequence of r^th-root of r^th moment for x=1/2, 2 starting at the second is, divided by, A , is : [0.2041241452, 0.2554364776, 0.3518576419, 0.4325703404, 0.5142587986, 0.5944150707, 0.6738464234, 0.7526727271, 0.8310337000] The whole thing took, 2.624, seconds of CPU time This took, 2.656, seconds of CPU time