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 the game The Expected value of D is -n (n - A) The , 2, -th moment (about the mean) of D is 2 2 n (n - A) (2 n - 2 A n + A - 2) - --------------------------------- 3 and setting n=, A x, it is 2 2 2 2 2 A x (x - 1) (2 A x - 2 A x + A - 2) - ---------------------------------------- 3 and asymptotically it is, as, A, goes to infinity 2 4 x (x - 1) (2 x - 2 x + 1) A - ----------------------------- 3 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 - n (n - A) 4 3 2 2 2 3 2 4 (16 n - 32 A n - 20 n + 28 n A + 20 A n - 12 n A + 4 - 10 A + 3 A )/ 15 and setting n=, A x, it is 2 4 4 4 3 2 2 4 2 2 4 - A x (x - 1) (16 A x - 32 A x - 20 A x + 28 A x + 20 A x - 12 A x 2 4 + 4 - 10 A + 3 A )/15 and asymptotically it is, as, A, goes to infinity 4 3 2 6 x (x - 1) (16 x - 32 x + 28 x - 12 x + 3) A - ----------------------------------------------- 15 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 5 4 4 2 3 3 3 - n (n - A) (132 n - 396 A n - 168 n + 528 n A + 336 A n - 396 n A 2 4 2 2 2 3 5 2 + 199 n A + 28 n - 364 n A + 196 n A - 28 A n - 67 n A + 84 A 6 4 + 17 A - 84 A + 8)/105 and setting n=, A x, it is 2 6 6 6 5 4 4 6 4 4 3 - A x (x - 1) (132 A x - 396 A x - 168 A x + 528 A x + 336 A x 6 3 6 2 2 2 4 2 4 2 - 396 A x + 199 A x + 28 A x - 364 A x + 196 A x - 28 A x 6 2 6 4 - 67 A x + 84 A + 17 A - 84 A + 8)/105 and asymptotically it is, as, A, goes to infinity 6 5 4 3 2 8 x (x - 1) (132 x - 396 x + 528 x - 396 x + 199 x - 67 x + 17) A - --------------------------------------------------------------------- 105 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 7 2 2 8 4 - n (n - A) (-144 - 4864 A n + 800 n - 440 A - 800 A n + 155 A - 672 n 3 4 3 8 6 2 2 2 + 1344 A n + 1764 A - 2016 n A + 1216 n + 8816 n A + 1344 n A 2 6 4 4 6 3 5 5 3 5 + 1790 n A + 6956 n A - 1200 n - 3880 n A - 9424 n A + 3600 A n 6 3 3 5 7 4 2 2 4 - 1020 A + 6960 n A + 2760 n A - 610 n A - 6480 n A - 5640 n A )/ 945 and setting n=, A x, it is 2 2 4 4 2 4 3 - A x (x - 1) (-144 - 800 A x - 2016 A x + 1344 A x + 1344 A x 4 4 2 8 7 8 6 8 2 8 4 - 672 A x - 440 A - 4864 A x + 8816 A x + 1790 A x + 6956 A x 8 3 8 5 8 8 4 2 2 - 3880 A x - 9424 A x - 610 A x + 155 A + 1764 A + 800 A x 6 6 6 6 5 6 4 6 3 6 2 - 1020 A - 1200 A x + 3600 A x - 6480 A x + 6960 A x - 5640 A x 6 8 8 + 2760 A x + 1216 A x )/945 and asymptotically it is, as, A, goes to infinity 7 6 2 4 3 5 - x (x - 1) (-4864 x + 8816 x + 1790 x + 6956 x - 3880 x - 9424 x - 610 x 8 10 + 155 + 1216 x ) A /945 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 7 2 2 10 - n (n - A) (-1440 + 12320 A n - 6160 n - 3344 A + 6160 A n + 2073 A 9 8 4 3 4 8 - 62200 A n - 17050 A + 31240 n - 62480 A n - 29040 A - 3080 n 6 2 2 2 2 6 4 4 6 - 55000 n A + 31240 n A - 113740 n A - 176110 n A - 33000 n 3 5 5 3 5 6 3 3 + 163460 n A + 121880 n A + 99000 A n + 42636 A - 13728 n A 5 7 4 2 2 4 4 6 - 73788 n A + 50270 n A - 75636 n A + 97152 n A + 93901 n A 6 4 2 8 8 2 3 7 9 + 193990 n A + 23908 n A + 143060 n A - 51332 n A - 8157 n A 7 3 5 5 10 - 199040 n A - 146570 n A + 12440 n )/10395 and setting n=, A x, it is 2 2 4 2 4 3 4 4 - A x (x - 1) (-1440 + 6160 A x + 31240 A x - 62480 A x + 31240 A x 2 10 8 7 8 6 8 2 - 3344 A + 2073 A + 12320 A x - 55000 A x - 113740 A x 8 4 8 3 8 5 8 8 - 176110 A x + 163460 A x + 121880 A x + 50270 A x - 17050 A 4 10 9 2 2 10 4 10 6 - 29040 A - 62200 A x - 6160 A x + 93901 A x + 193990 A x 10 2 10 10 10 8 10 3 10 + 23908 A x + 12440 A x + 143060 A x - 51332 A x - 8157 A x 10 7 10 5 6 6 6 6 5 - 199040 A x - 146570 A x + 42636 A - 33000 A x + 99000 A x 6 4 6 3 6 2 6 8 8 - 75636 A x - 13728 A x + 97152 A x - 73788 A x - 3080 A x )/10395 and asymptotically it is, as, A, goes to infinity 9 4 6 2 10 - x (x - 1) (2073 - 62200 x + 93901 x + 193990 x + 23908 x + 12440 x 8 3 7 5 12 + 143060 x - 51332 x - 8157 x - 199040 x - 146570 x ) A /10395 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 7 2 2 - n (n - A) (46656 + 3907904 A n - 215488 n + 84448 A + 215488 A n 10 9 8 4 3 - 377286 A - 851760 A n + 1241240 A - 48048 n + 96096 A n 4 3 8 6 2 2 2 + 240240 A + 544544 n A - 976976 n - 6086080 n A - 592592 n A 12 2 6 4 4 12 6 + 38227 A + 4496492 n A + 208208 n A + 138048 n + 885456 n 3 5 5 3 5 6 - 3491488 n A + 4580576 n A - 2656368 A n - 1439152 A 3 3 5 7 4 2 - 2512224 n A + 1203488 n A - 2638636 n A + 3469752 n A 2 4 10 2 8 4 6 6 - 390104 n A + 2312304 n A + 4819900 n A + 3710792 n A 2 10 4 8 4 6 6 4 + 440811 n A + 1718087 n A - 5738642 n A - 2513420 n A 2 8 8 2 3 7 9 - 2819726 n A + 1350440 n A + 4537624 n A + 1174264 n A 7 3 5 5 10 11 3 9 - 291200 n A + 4982068 n A + 170352 n - 828288 A n - 945574 n A 5 7 7 5 9 3 11 - 2669296 n A - 4577488 n A - 3968880 n A - 150416 n A )/135135 and setting n=, A x, it is 2 2 4 4 2 4 3 - A x (x - 1) (46656 + 215488 A x + 544544 A x - 592592 A x + 96096 A x 4 4 2 10 8 7 8 6 - 48048 A x + 84448 A - 377286 A + 3907904 A x - 6086080 A x 8 2 8 4 8 3 8 5 + 4496492 A x + 208208 A x - 3491488 A x + 4580576 A x 8 8 4 10 9 12 - 2638636 A x + 1241240 A + 240240 A - 851760 A x + 38227 A 2 2 10 4 10 6 10 2 - 215488 A x - 5738642 A x - 2513420 A x - 2819726 A x 10 10 10 8 10 3 10 + 170352 A x + 1350440 A x + 4537624 A x + 1174264 A x 10 7 10 5 12 10 6 - 291200 A x + 4982068 A x + 2312304 A x - 1439152 A 12 12 6 6 6 5 6 4 + 138048 A x + 885456 A x - 2656368 A x + 3469752 A x 6 3 6 2 6 8 8 - 2512224 A x - 390104 A x + 1203488 A x - 976976 A x 12 8 12 6 12 2 12 4 + 4819900 A x + 3710792 A x + 440811 A x + 1718087 A x 12 11 12 3 12 5 12 7 - 828288 A x - 945574 A x - 2669296 A x - 4577488 A x 12 9 12 - 3968880 A x - 150416 A x)/135135 and asymptotically it is, as, A, goes to infinity 10 12 8 6 - x (x - 1) (38227 + 2312304 x + 138048 x + 4819900 x + 3710792 x 2 4 11 3 5 + 440811 x + 1718087 x - 828288 x - 945574 x - 2669296 x 7 9 14 - 4577488 x - 3968880 x - 150416 x) A /135135 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 3 11 11 3 9 5 - n (n - A) (1088640 - 22990971 n A - 75259408 n A - 125838664 n A 7 14 13 5 9 - 82001920 A n + 1601264 n - 3657671 n A - 64559589 n A 7 7 2 13 2 - 109462304 n A + 3385536 n - 11208848 A n + 2213184 A - 3385536 A n 10 9 8 4 + 43765176 A + 126832160 A n - 72701200 A - 16278080 n 3 4 3 8 + 32556160 A n + 5154240 A - 4979520 n A + 20500480 n 6 2 2 2 12 2 6 + 172446560 n A - 11298560 n A - 10703560 A - 111654400 n A 4 4 12 2 12 6 + 157174160 n A + 6628160 n + 10719069 n A + 8440432 n 3 5 5 3 5 6 - 26128960 n A - 230332960 n A - 25321296 A n + 36212176 A 3 3 5 7 4 2 + 60172112 n A + 7383376 n A + 99997040 n A - 8984976 n A 2 4 10 2 8 4 6 6 - 41689648 n A + 99701280 n A + 81057200 n A - 163707320 n A 12 2 8 6 4 10 6 8 + 36829072 n A + 123650576 n A + 41746611 n A + 88716391 n A 10 4 2 10 4 8 14 + 109714472 n A - 86189880 n A - 207190760 n A + 929569 A 4 6 6 4 2 8 8 2 + 161178472 n A - 250785080 n A + 204565816 n A - 279173440 n A 3 7 9 7 3 5 5 - 227866184 n A - 105183624 n A + 355700800 n A + 40097512 n A 10 11 3 9 5 7 - 25366432 n - 39768960 A n + 148044120 n A + 218754760 n A 7 5 9 3 11 + 42058240 n A - 133957600 n A + 34570760 n A )/2027025 and setting n=, A x, it is 2 2 4 4 2 - A x (x - 1) (1088640 - 3385536 A x - 4979520 A x - 11298560 A x 4 3 4 4 2 10 + 32556160 A x - 16278080 A x + 2213184 A + 43765176 A 8 7 8 6 8 2 8 4 - 82001920 A x + 172446560 A x - 111654400 A x + 157174160 A x 8 3 8 5 8 8 - 26128960 A x - 230332960 A x + 99997040 A x - 72701200 A 4 10 9 12 2 2 + 5154240 A + 126832160 A x - 10703560 A + 3385536 A x 10 4 10 6 10 2 + 161178472 A x - 250785080 A x + 204565816 A x 10 10 10 8 10 3 10 - 25366432 A x - 279173440 A x - 227866184 A x - 105183624 A x 10 7 10 5 12 10 14 14 + 355700800 A x + 40097512 A x + 99701280 A x + 1601264 A x 6 14 3 14 11 14 9 + 36212176 A - 22990971 A x - 75259408 A x - 125838664 A x 14 14 5 14 7 14 13 - 3657671 A x - 64559589 A x - 109462304 A x - 11208848 A x 14 2 12 12 14 12 6 6 + 10719069 A x + 6628160 A x + 36829072 A x + 8440432 A x 6 5 6 4 6 3 6 2 - 25321296 A x - 8984976 A x + 60172112 A x - 41689648 A x 6 8 8 12 8 12 6 + 7383376 A x + 20500480 A x + 81057200 A x - 163707320 A x 12 2 12 4 12 11 12 3 - 86189880 A x - 207190760 A x - 39768960 A x + 148044120 A x 12 5 12 7 12 9 12 + 218754760 A x + 42058240 A x - 133957600 A x + 34570760 A x 14 8 14 4 14 6 14 10 + 123650576 A x + 41746611 A x + 88716391 A x + 109714472 A x 14 + 929569 A )/2027025 and asymptotically it is, as, A, goes to infinity 14 3 11 9 - x (x - 1) (1601264 x - 22990971 x - 75259408 x - 125838664 x 5 7 13 2 - 3657671 x - 64559589 x - 109462304 x - 11208848 x + 10719069 x 12 8 4 6 10 + 36829072 x + 123650576 x + 41746611 x + 88716391 x + 109714472 x 16 + 929569) A /2027025 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 3 11 11 3 - n (n - A) (-56367360 + 5762006928 n A - 7735007616 n A 9 5 7 14 13 - 6169839648 n A - 2804246016 A n + 191538048 n + 1258380528 n A 5 9 7 7 2 13 + 10038706992 n A + 5614481472 n A + 258665472 n - 1340766336 A n 2 10 9 - 92103552 A - 258665472 A n - 4038900528 A - 1821322880 A n 8 4 3 4 + 3454548240 A + 43034752 n - 86069504 A n - 139950528 A 3 8 6 2 15 - 478567680 n A + 701061504 n + 2802379008 n A - 144871424 A n 2 2 12 2 6 4 4 + 521602432 n A + 1855997304 A - 23337600 n A - 5620160832 n A 12 2 12 6 3 5 - 628045824 n - 3218884992 n A - 892260096 n + 5622494592 n A 5 3 4 12 5 6 + 1407724032 n A + 1294216554 n A + 2676780288 A n - 358507968 A 3 13 5 11 3 3 - 712809600 n A - 2000550012 n A + 1172056704 n A 5 7 14 2 10 6 - 1147106688 n A - 2085914688 n A + 547795072 n A + 3510262672 n A 6 10 8 8 12 4 + 2740748628 n A + 3696083008 n A + 2216820256 n A 2 14 4 2 16 2 4 + 332336460 n A - 2816678592 n A + 18108928 n + 1007208384 n A 10 2 8 4 6 6 16 - 10549469952 n A - 20855311680 n A - 4418523312 n A + 28820619 A 12 2 8 6 4 10 + 4194161664 n A + 274838592 n A - 8547941088 n A 6 8 10 4 2 10 - 9345503568 n A + 9023829024 n A + 10197582192 n A 4 8 14 4 6 6 4 + 13413386784 n A - 379264152 A + 2891277584 n A + 15804788480 n A 2 8 8 2 3 7 - 10606490128 n A + 5733058240 n A + 5603223392 n A 9 7 3 5 5 10 + 7084383072 n A - 12004295680 n A - 13048886656 n A + 364264576 n 7 9 9 7 11 5 - 3342926112 n A - 3741026672 n A - 3001468736 n A 13 3 15 11 - 1299315584 n A - 113403438 n A + 3768274944 A n 3 9 5 7 7 5 - 13449241008 n A - 6486701424 n A + 15643294464 n A 9 3 11 + 18204829440 n A - 4840074624 n A )/34459425 and setting n=, A x, it is 2 2 4 4 2 - A x (x - 1) (-56367360 - 258665472 A x - 478567680 A x + 521602432 A x 4 3 4 4 2 10 - 86069504 A x + 43034752 A x - 92103552 A - 4038900528 A 8 7 8 6 8 2 8 4 - 2804246016 A x + 2802379008 A x - 23337600 A x - 5620160832 A x 8 3 8 5 8 8 + 5622494592 A x + 1407724032 A x - 2085914688 A x + 3454548240 A 4 10 9 12 2 2 - 139950528 A - 1821322880 A x + 1855997304 A + 258665472 A x 10 4 10 6 10 2 + 2891277584 A x + 15804788480 A x - 10606490128 A x 10 10 10 8 10 3 + 364264576 A x + 5733058240 A x + 5603223392 A x 10 10 7 10 5 + 7084383072 A x - 12004295680 A x - 13048886656 A x 12 10 16 15 16 4 - 10549469952 A x - 144871424 A x + 1294216554 A x 16 3 14 14 6 14 3 - 712809600 A x + 191538048 A x - 358507968 A + 5762006928 A x 14 11 14 9 14 - 7735007616 A x - 6169839648 A x + 1258380528 A x 16 7 16 9 16 11 - 3342926112 A x - 3741026672 A x - 3001468736 A x 16 13 16 14 5 - 1299315584 A x - 113403438 A x + 10038706992 A x 14 7 14 13 14 2 + 5614481472 A x - 1340766336 A x - 3218884992 A x 12 12 16 14 12 6 6 - 628045824 A x + 28820619 A + 4194161664 A x - 892260096 A x 6 5 6 4 6 3 + 2676780288 A x - 2816678592 A x + 1172056704 A x 6 2 6 8 8 + 1007208384 A x - 1147106688 A x + 701061504 A x 12 8 12 6 12 2 - 20855311680 A x - 4418523312 A x + 10197582192 A x 12 4 12 11 12 3 + 13413386784 A x + 3768274944 A x - 13449241008 A x 12 5 12 7 12 9 - 6486701424 A x + 15643294464 A x + 18204829440 A x 12 16 16 14 8 - 4840074624 A x + 18108928 A x + 274838592 A x 14 4 14 6 14 10 - 8547941088 A x - 9345503568 A x + 9023829024 A x 14 16 5 16 14 - 379264152 A - 2000550012 A x + 547795072 A x 16 10 16 6 16 8 + 3510262672 A x + 2740748628 A x + 3696083008 A x 16 12 16 2 + 2216820256 A x + 332336460 A x )/34459425 and asymptotically it is, as, A, goes to infinity 15 4 3 7 - x (x - 1) (-144871424 x + 1294216554 x - 712809600 x - 3342926112 x 9 11 13 - 3741026672 x - 3001468736 x - 1299315584 x - 113403438 x + 28820619 16 5 14 10 + 18108928 x - 2000550012 x + 547795072 x + 3510262672 x 6 8 12 2 18 + 2740748628 x + 3696083008 x + 2216820256 x + 332336460 x ) A / 34459425 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 3 11 11 3 - n (n - A) (-2358028800 - 838130410416 n A + 778469607552 n A 9 5 7 14 + 1335000431856 n A + 159669494400 A n - 15196462656 n 13 5 9 7 7 - 258961242216 n A - 758447633424 n A + 466251854016 n A 2 13 2 17 - 6189532416 n + 106375238592 A n - 4520265984 A - 4366264705 n A 10 9 8 + 6189532416 A n + 310438250928 A - 193854729120 A n - 116527288800 A 4 18 3 4 + 30285513600 n + 1109652905 A - 60571027200 A n - 9112631040 A 3 8 6 2 + 8387043840 n A - 39917373600 n - 281846285760 n A 15 2 2 12 - 41553273600 A n + 21898469760 n A - 252873134080 A 2 6 4 4 12 + 150095774400 n A - 65258506560 n A + 2026372800 n 17 5 13 7 11 - 1457411616 A n - 77020838014 n A - 128409398344 n A 9 9 11 7 13 5 - 138411206394 n A - 99867787584 n A - 53051056464 n A 15 3 3 15 2 12 - 16841200896 n A - 27444592570 n A + 575710512624 n A 6 3 5 5 3 - 12777528576 n - 161027954880 n A + 286695626880 n A 4 12 5 6 - 411053148180 n A + 38332585728 A n - 19190210368 A 3 13 5 11 3 3 + 266990861400 n A + 513006337260 n A - 91194900096 n A 5 7 14 2 + 7400389952 n A - 48410774880 n A + 153244253760 n A 10 6 6 10 8 8 + 416701012560 n A - 532119639960 n A - 186415760010 n A 12 4 2 14 4 2 + 528809351280 n A - 144981426600 n A + 13653628608 n A 16 2 4 10 2 + 5194159200 n + 44585824384 n A + 105128903040 n A 8 4 6 6 16 + 903504256400 n A + 1094364594640 n A - 16427752830 A 12 2 8 6 4 10 - 360224618208 n A - 1064819270424 n A + 976477718736 n A 6 8 10 4 2 10 + 244633336896 n A - 1187139062928 n A - 893624745840 n A 4 8 14 4 6 - 361276197680 n A + 93678245544 A - 579984422784 n A 6 4 2 8 8 2 - 209551598880 n A + 326124035328 n A + 349472020560 n A 3 7 9 7 3 + 213585460608 n A - 345925914672 n A - 234759707520 n A 5 5 10 2 16 + 636123910656 n A + 38770945824 n + 12795632450 n A 4 14 6 12 8 10 + 49829537246 n A + 105486658856 n A + 140665272001 n A 10 8 12 6 14 4 + 123080594826 n A + 75451250616 n A + 33164392944 n A 16 2 18 7 9 + 6234483024 n A + 161934624 n + 420223637640 n A 9 7 11 5 13 3 - 126478604160 n A - 571672708320 n A - 345527488320 n A 15 11 3 9 + 55632436050 n A - 12158236800 A n + 794195216560 n A 5 7 7 5 9 3 - 469890554960 n A - 1262593563200 n A - 414194011200 n A 11 + 514517966240 n A )/654729075 and setting n=, A x, it is 2 2 4 - A x (x - 1) (6189532416 A x - 2358028800 + 8387043840 A x 4 2 4 3 4 4 + 21898469760 A x - 60571027200 A x + 30285513600 A x 2 18 18 18 2 - 4520265984 A + 161934624 A x + 12795632450 A x 18 4 18 6 18 8 + 49829537246 A x + 105486658856 A x + 140665272001 A x 18 10 18 12 18 14 + 123080594826 A x + 75451250616 A x + 33164392944 A x 18 16 10 8 7 + 6234483024 A x + 310438250928 A + 159669494400 A x 8 6 8 2 8 4 - 281846285760 A x + 150095774400 A x - 65258506560 A x 8 3 8 5 8 - 161027954880 A x + 286695626880 A x - 48410774880 A x 8 18 4 10 9 - 116527288800 A + 1109652905 A - 9112631040 A - 193854729120 A x 12 2 2 10 4 - 252873134080 A - 6189532416 A x - 579984422784 A x 10 6 10 2 10 10 - 209551598880 A x + 326124035328 A x + 38770945824 A x 10 8 10 3 10 + 349472020560 A x + 213585460608 A x - 345925914672 A x 10 7 10 5 12 10 - 234759707520 A x + 636123910656 A x + 105128903040 A x 16 15 16 4 16 3 - 41553273600 A x - 411053148180 A x + 266990861400 A x 14 14 6 14 3 - 15196462656 A x - 19190210368 A - 838130410416 A x 14 11 14 9 14 + 778469607552 A x + 1335000431856 A x - 258961242216 A x 16 7 16 9 16 11 + 420223637640 A x - 126478604160 A x - 571672708320 A x 16 13 16 14 5 - 345527488320 A x + 55632436050 A x - 758447633424 A x 14 7 14 13 14 2 + 466251854016 A x + 106375238592 A x + 575710512624 A x 12 12 16 14 12 + 2026372800 A x - 16427752830 A - 360224618208 A x 6 6 6 5 6 4 - 12777528576 A x + 38332585728 A x + 13653628608 A x 6 3 6 2 6 - 91194900096 A x + 44585824384 A x + 7400389952 A x 8 8 18 12 8 - 39917373600 A x - 4366264705 A x + 903504256400 A x 12 6 12 2 12 4 + 1094364594640 A x - 893624745840 A x - 361276197680 A x 12 11 12 3 12 5 - 12158236800 A x + 794195216560 A x - 469890554960 A x 12 7 12 9 12 - 1262593563200 A x - 414194011200 A x + 514517966240 A x 16 16 18 17 18 5 + 5194159200 A x - 1457411616 A x - 77020838014 A x 18 7 18 9 18 11 - 128409398344 A x - 138411206394 A x - 99867787584 A x 18 13 18 15 14 8 - 53051056464 A x - 16841200896 A x - 1064819270424 A x 14 4 14 6 14 10 + 976477718736 A x + 244633336896 A x - 1187139062928 A x 14 18 3 16 5 + 93678245544 A - 27444592570 A x + 513006337260 A x 16 14 16 10 16 6 + 153244253760 A x + 416701012560 A x - 532119639960 A x 16 8 16 12 16 2 - 186415760010 A x + 528809351280 A x - 144981426600 A x )/ 654729075 and asymptotically it is, as, A, goes to infinity 18 2 4 6 - x (x - 1) (161934624 x + 12795632450 x + 49829537246 x + 105486658856 x 8 10 12 14 + 140665272001 x + 123080594826 x + 75451250616 x + 33164392944 x 16 17 + 6234483024 x + 1109652905 - 4366264705 x - 1457411616 x 5 7 9 11 - 77020838014 x - 128409398344 x - 138411206394 x - 99867787584 x 13 15 3 20 - 53051056464 x - 16841200896 x - 27444592570 x ) A /654729075 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 3 2 2\1/2 | 3 (16 x - 32 x + 28 x - 12 x + 3) | |- -------------------------------------| | 2 3 | \ x (x - 1) (2 x - 2 x + 1) / -------------------------------------------- 5 and when, x = 1/2, it is 1/2 96 ----- 5 which is roughly, 1.959591794 The asymptotic Kurtosis is 4 3 2 3 (66 x - 132 x + 99 x - 33 x + 17) - -------------------------------------- 2 35 (2 x - 2 x + 1) (x - 1) x 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, 2.384, seconds of CPU time This took, 2.416, seconds of CPU time