The Expectation, Variance, and first, 12, moments of the random variable: rea\ ching n Heads or n Tails By Tossing a Fair Coin By Shalosh B. Ekhad binomial(2 n, n + 1) (n + 1) Let C denote , ---------------------------- n 4 / n \1/2 Note that it is asymptotically, |----| , plus O(1/sqrt(n)) \ Pi / You toss a fair coin and stop as soon as you have reached n Heads OR n Tail\ s, How long should it take? The expectation of this random variable is 2 n - 2 C and in Maple notation 2*n-2*C The variance of this random variable is 2 -4 C - 2 C + 2 n and in Maple notation -4*C^2-2*C+2*n Hence the limit of the coefficient of variation is, 0 The , 3, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 3 2 -16 C - 12 C + 4 C n - 6 C + 6 n and in Maple notation -16*C^3-12*C^2+4*C*n-6*C+6*n and the limit of the, 3, scaled moment is 1/2 (Pi - 4) 2 ------------- 3/2 (Pi - 2) and in Maple notation (Pi-4)*2^(1/2)/(Pi-2)^(3/2) and in deminals -.9952717461 The , 4, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 4 3 2 2 2 -48 C - 48 C - 16 C n - 48 C + 12 n - 26 C + 26 n and in Maple notation -48*C^4-48*C^3-16*C^2*n-48*C^2+12*n^2-26*C+26*n and the limit of the, 4, scaled moment is 2 3 Pi - 4 Pi - 12 ----------------- 2 (Pi - 2) and in Maple notation (3*Pi^2-4*Pi-12)/(Pi-2)^2 and in deminals 3.869177308 The , 5, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 5 4 3 3 2 2 2 -128 C - 160 C - 160 C n - 240 C - 240 C n + 56 C n - 260 C - 68 C n 2 + 120 n - 150 C + 150 n and in Maple notation -128*C^5-160*C^4-160*C^3*n-240*C^3-240*C^2*n+56*C*n^2-260*C^2-68*C*n+120*n^2-\ 150*C+150*n and the limit of the, 5, scaled moment is 2 1/2 (7 Pi - 20 Pi - 16) 2 ------------------------- 5/2 (Pi - 2) and in Maple notation (7*Pi^2-20*Pi-16)*2^(1/2)/(Pi-2)^(5/2) and in deminals -9.896966340 The , 6, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 6 5 4 4 3 2 2 3 -320 C - 480 C - 800 C n - 960 C - 1920 C n - 48 C n - 1560 C 2 2 3 2 2 - 2376 C n + 480 C n + 120 n - 1800 C - 840 C n + 1140 n - 1082 C + 1082 n and in Maple notation -320*C^6-480*C^5-800*C^4*n-960*C^4-1920*C^3*n-48*C^2*n^2-1560*C^3-2376*C^2*n+ 480*C*n^2+120*n^3-1800*C^2-840*C*n+1140*n^2-1082*C+1082*n and the limit of the, 6, scaled moment is 3 2 15 Pi - 6 Pi - 100 Pi - 40 ---------------------------- 3 (Pi - 2) and in Maple notation (15*Pi^3-6*Pi^2-100*Pi-40)/(Pi-2)^3 and in deminals 34.76177935 The , 7, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 7 6 5 5 4 3 2 4 -768 C - 1344 C - 3136 C n - 3360 C - 10080 C n - 2016 C n - 7280 C 3 2 2 3 3 2 2 - 20272 C n - 3360 C n + 912 C n - 12600 C - 24360 C n + 3544 C n 3 2 2 + 2520 n - 15148 C - 9596 C n + 11760 n - 9366 C + 9366 n and in Maple notation -768*C^7-1344*C^6-3136*C^5*n-3360*C^5-10080*C^4*n-2016*C^3*n^2-7280*C^4-20272*C ^3*n-3360*C^2*n^2+912*C*n^3-12600*C^3-24360*C^2*n+3544*C*n^2+2520*n^3-15148*C^2 -9596*C*n+11760*n^2-9366*C+9366*n and the limit of the, 7, scaled moment is 3 2 1/2 (57 Pi - 126 Pi - 196 Pi - 48) 2 ------------------------------------- 7/2 (Pi - 2) and in Maple notation (57*Pi^3-126*Pi^2-196*Pi-48)*2^(1/2)/(Pi-2)^(7/2) and in deminals -124.5213875 The , 8, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 8 7 6 6 5 4 2 -1792 C - 3584 C - 10752 C n - 10752 C - 43008 C n - 15232 C n 5 4 3 2 2 3 4 - 29120 C - 117824 C n - 53760 C n + 1152 C n - 67200 C 3 2 2 3 4 3 - 228480 C n - 70976 C n + 18816 C n + 1680 n - 121184 C 2 2 3 2 2 - 274720 C n + 24192 C n + 42000 n - 149856 C - 115920 C n + 134652 n - 94586 C + 94586 n and in Maple notation -1792*C^8-3584*C^7-10752*C^6*n-10752*C^6-43008*C^5*n-15232*C^4*n^2-29120*C^5-\ 117824*C^4*n-53760*C^3*n^2+1152*C^2*n^3-67200*C^4-228480*C^3*n-70976*C^2*n^2+ 18816*C*n^3+1680*n^4-121184*C^3-274720*C^2*n+24192*C*n^2+42000*n^3-149856*C^2-\ 115920*C*n+134652*n^2-94586*C+94586*n and the limit of the, 8, scaled moment is 4 3 2 105 Pi + 72 Pi - 952 Pi - 672 Pi - 112 ----------------------------------------- 4 (Pi - 2) and in Maple notation (105*Pi^4+72*Pi^3-952*Pi^2-672*Pi-112)/(Pi-2)^4 and in deminals 495.3978844 The , 9, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 9 8 7 7 6 5 2 -4096 C - 9216 C - 33792 C n - 32256 C - 161280 C n - 80640 C n 6 5 4 2 3 3 5 - 104832 C - 556416 C n - 403200 C n - 29952 C n - 302400 C 4 3 2 2 3 4 4 - 1471680 C n - 1021824 C n - 24192 C n + 17952 C n - 727104 C 3 2 2 3 4 3 - 2836032 C n - 1257984 C n + 302880 C n + 60480 n - 1348704 C 2 2 3 2 - 3435264 C n + 118008 C n + 680400 n - 1702548 C - 1521908 C n 2 + 1711080 n - 1091670 C + 1091670 n and in Maple notation -4096*C^9-9216*C^8-33792*C^7*n-32256*C^7-161280*C^6*n-80640*C^5*n^2-104832*C^6-\ 556416*C^5*n-403200*C^4*n^2-29952*C^3*n^3-302400*C^5-1471680*C^4*n-1021824*C^3* n^2-24192*C^2*n^3+17952*C*n^4-727104*C^4-2836032*C^3*n-1257984*C^2*n^2+302880*C *n^3+60480*n^4-1348704*C^3-3435264*C^2*n+118008*C*n^2+680400*n^3-1702548*C^2-\ 1521908*C*n+1711080*n^2-1091670*C+1091670*n and the limit of the, 9, scaled moment is 1/2 4 3 2 2 (561 Pi - 936 Pi - 2520 Pi - 1056 Pi - 128) --------------------------------------------------- 9/2 (Pi - 2) and in Maple notation 2^(1/2)*(561*Pi^4-936*Pi^3-2520*Pi^2-1056*Pi-128)/(Pi-2)^(9/2) and in deminals -2098.157939 The , 10, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 10 9 8 8 7 6 2 -9216 C - 23040 C - 99840 C n - 92160 C - 552960 C n - 354816 C n 7 6 5 2 4 3 6 - 349440 C - 2295552 C n - 2257920 C n - 334080 C n - 1209600 C 5 4 2 3 3 2 4 5 - 7660800 C n - 8088960 C n - 1451520 C n + 56640 C n - 3635520 C 4 3 2 2 3 4 5 - 20118720 C n - 18224640 C n - 1502400 C n + 656640 C n + 30240 n 4 3 2 2 3 4 - 8991360 C - 38848320 C n - 21877200 C n + 4668480 C n + 1562400 n 3 2 2 3 2 - 17025480 C - 47463640 C n - 709920 C n + 11357640 n - 21833400 C 2 - 21815640 C n + 23998980 n - 14174522 C + 14174522 n and in Maple notation -9216*C^10-23040*C^9-99840*C^8*n-92160*C^8-552960*C^7*n-354816*C^6*n^2-349440*C ^7-2295552*C^6*n-2257920*C^5*n^2-334080*C^4*n^3-1209600*C^6-7660800*C^5*n-\ 8088960*C^4*n^2-1451520*C^3*n^3+56640*C^2*n^4-3635520*C^5-20118720*C^4*n-\ 18224640*C^3*n^2-1502400*C^2*n^3+656640*C*n^4+30240*n^5-8991360*C^4-38848320*C^ 3*n-21877200*C^2*n^2+4668480*C*n^3+1562400*n^4-17025480*C^3-47463640*C^2*n-\ 709920*C*n^2+11357640*n^3-21833400*C^2-21815640*C*n+23998980*n^2-14174522*C+ 14174522*n and the limit of the, 10, scaled moment is 5 4 3 2 945 Pi + 1770 Pi - 10440 Pi - 11088 Pi - 3120 Pi - 288 ---------------------------------------------------------- 5 (Pi - 2) and in Maple notation (945*Pi^5+1770*Pi^4-10440*Pi^3-11088*Pi^2-3120*Pi-288)/(Pi-2)^5 and in deminals 9476.097016 The , 11, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 11 10 9 9 8 7 2 -20480 C - 56320 C - 281600 C n - 253440 C - 1774080 C n - 1385472 C n 8 7 6 2 5 3 7 - 1098240 C - 8600064 C n - 10644480 C n - 2280960 C n - 4435200 C 6 5 2 4 3 3 4 - 34594560 C n - 48702720 C n - 15079680 C n - 485760 C n 6 5 4 2 3 3 - 15996288 C - 114652032 C n - 154344960 C n - 44246400 C n 2 4 5 5 4 + 1140480 C n + 419520 C n - 49452480 C - 301371840 C n 3 2 2 3 4 5 - 329519520 C n - 46981440 C n + 17108160 C n + 1663200 n 4 3 2 2 3 - 124853520 C - 584526800 C n - 392055840 C n + 73053744 C n 4 3 2 2 + 36590400 n - 240167400 C - 720111480 C n - 43277992 C n 3 2 2 + 199500840 n - 311839484 C - 340717676 C n + 369066720 n - 204495126 C + 204495126 n and in Maple notation -20480*C^11-56320*C^10-281600*C^9*n-253440*C^9-1774080*C^8*n-1385472*C^7*n^2-\ 1098240*C^8-8600064*C^7*n-10644480*C^6*n^2-2280960*C^5*n^3-4435200*C^7-34594560 *C^6*n-48702720*C^5*n^2-15079680*C^4*n^3-485760*C^3*n^4-15996288*C^6-114652032* C^5*n-154344960*C^4*n^2-44246400*C^3*n^3+1140480*C^2*n^4+419520*C*n^5-49452480* C^5-301371840*C^4*n-329519520*C^3*n^2-46981440*C^2*n^3+17108160*C*n^4+1663200*n ^5-124853520*C^4-584526800*C^3*n-392055840*C^2*n^2+73053744*C*n^3+36590400*n^4-\ 240167400*C^3-720111480*C^2*n-43277992*C*n^2+199500840*n^3-311839484*C^2-\ 340717676*C*n+369066720*n^2-204495126*C+204495126*n and the limit of the, 11, scaled moment is 5 4 3 2 1/2 (6555 Pi - 7590 Pi - 35640 Pi - 21648 Pi - 4400 Pi - 320) 2 ------------------------------------------------------------------ 11/2 (Pi - 2) and in Maple notation (6555*Pi^5-7590*Pi^4-35640*Pi^3-21648*Pi^2-4400*Pi-320)*2^(1/2)/(Pi-2)^(11/2) and in deminals -45219.22019 The , 12, -th central limit is , in terms of C= binomial(2*n,n+1)*(n+1)/4^n, is: 12 11 10 10 9 -45056 C - 135168 C - 765952 C n - 675840 C - 5406720 C n 8 2 9 8 7 2 - 4967424 C n - 3294720 C - 29956608 C n - 44605440 C n 6 3 8 7 6 2 - 12367872 C n - 15206400 C - 140912640 C n - 247256064 C n 5 3 4 4 7 6 - 106444800 C n - 8321280 C n - 63985152 C - 563126784 C n 5 2 4 3 3 4 2 5 - 1000581120 C n - 464851200 C n - 39536640 C n + 2085120 C n 6 5 4 2 3 3 - 237371904 C - 1867574016 C n - 2991637440 C n - 1162529280 C n 2 4 5 6 5 - 1877760 C n + 23696640 C n + 665280 n - 749121120 C 4 3 2 2 3 4 - 4925921440 C n - 6210420480 C n - 1245127104 C n + 401702400 C n 5 4 3 2 2 + 61538400 n - 1921339200 C - 9602007360 C n - 7374402528 C n 3 4 3 2 + 1183799232 C n + 839306160 n - 3742073808 C - 11919298032 C n 2 3 2 - 1209981696 C n + 3716090400 n - 4907883024 C - 5773928160 C n 2 + 6182918412 n - 3245265146 C + 3245265146 n and in Maple notation -45056*C^12-135168*C^11-765952*C^10*n-675840*C^10-5406720*C^9*n-4967424*C^8*n^2 -3294720*C^9-29956608*C^8*n-44605440*C^7*n^2-12367872*C^6*n^3-15206400*C^8-\ 140912640*C^7*n-247256064*C^6*n^2-106444800*C^5*n^3-8321280*C^4*n^4-63985152*C^ 7-563126784*C^6*n-1000581120*C^5*n^2-464851200*C^4*n^3-39536640*C^3*n^4+2085120 *C^2*n^5-237371904*C^6-1867574016*C^5*n-2991637440*C^4*n^2-1162529280*C^3*n^3-\ 1877760*C^2*n^4+23696640*C*n^5+665280*n^6-749121120*C^5-4925921440*C^4*n-\ 6210420480*C^3*n^2-1245127104*C^2*n^3+401702400*C*n^4+61538400*n^5-1921339200*C ^4-9602007360*C^3*n-7374402528*C^2*n^2+1183799232*C*n^3+839306160*n^4-\ 3742073808*C^3-11919298032*C^2*n-1209981696*C*n^2+3716090400*n^3-4907883024*C^2 -5773928160*C*n+6182918412*n^2-3245265146*C+3245265146*n and the limit of the, 12, scaled moment is 6 5 4 3 2 (10395 Pi + 32580 Pi - 130020 Pi - 193248 Pi - 77616 Pi - 11968 Pi - 704) / 6 / (Pi - 2) / and in Maple notation (10395*Pi^6+32580*Pi^5-130020*Pi^4-193248*Pi^3-77616*Pi^2-11968*Pi-704)/(Pi-2)^ 6 and in deminals 226972.5454 ----------------------------------- To sum up the first, 12, moments are [2*n-2*C, -4*C^2-2*C+2*n, -16*C^3-12*C^2+4*C*n-6*C+6*n, -48*C^4-48*C^3-16*C^2*n -48*C^2+12*n^2-26*C+26*n, -128*C^5-160*C^4-160*C^3*n-240*C^3-240*C^2*n+56*C*n^2 -260*C^2-68*C*n+120*n^2-150*C+150*n, -320*C^6-480*C^5-800*C^4*n-960*C^4-1920*C^ 3*n-48*C^2*n^2-1560*C^3-2376*C^2*n+480*C*n^2+120*n^3-1800*C^2-840*C*n+1140*n^2-\ 1082*C+1082*n, -768*C^7-1344*C^6-3136*C^5*n-3360*C^5-10080*C^4*n-2016*C^3*n^2-\ 7280*C^4-20272*C^3*n-3360*C^2*n^2+912*C*n^3-12600*C^3-24360*C^2*n+3544*C*n^2+ 2520*n^3-15148*C^2-9596*C*n+11760*n^2-9366*C+9366*n, -1792*C^8-3584*C^7-10752*C ^6*n-10752*C^6-43008*C^5*n-15232*C^4*n^2-29120*C^5-117824*C^4*n-53760*C^3*n^2+ 1152*C^2*n^3-67200*C^4-228480*C^3*n-70976*C^2*n^2+18816*C*n^3+1680*n^4-121184*C ^3-274720*C^2*n+24192*C*n^2+42000*n^3-149856*C^2-115920*C*n+134652*n^2-94586*C+ 94586*n, -4096*C^9-9216*C^8-33792*C^7*n-32256*C^7-161280*C^6*n-80640*C^5*n^2-\ 104832*C^6-556416*C^5*n-403200*C^4*n^2-29952*C^3*n^3-302400*C^5-1471680*C^4*n-\ 1021824*C^3*n^2-24192*C^2*n^3+17952*C*n^4-727104*C^4-2836032*C^3*n-1257984*C^2* n^2+302880*C*n^3+60480*n^4-1348704*C^3-3435264*C^2*n+118008*C*n^2+680400*n^3-\ 1702548*C^2-1521908*C*n+1711080*n^2-1091670*C+1091670*n, -9216*C^10-23040*C^9-\ 99840*C^8*n-92160*C^8-552960*C^7*n-354816*C^6*n^2-349440*C^7-2295552*C^6*n-\ 2257920*C^5*n^2-334080*C^4*n^3-1209600*C^6-7660800*C^5*n-8088960*C^4*n^2-\ 1451520*C^3*n^3+56640*C^2*n^4-3635520*C^5-20118720*C^4*n-18224640*C^3*n^2-\ 1502400*C^2*n^3+656640*C*n^4+30240*n^5-8991360*C^4-38848320*C^3*n-21877200*C^2* n^2+4668480*C*n^3+1562400*n^4-17025480*C^3-47463640*C^2*n-709920*C*n^2+11357640 *n^3-21833400*C^2-21815640*C*n+23998980*n^2-14174522*C+14174522*n, -20480*C^11-\ 56320*C^10-281600*C^9*n-253440*C^9-1774080*C^8*n-1385472*C^7*n^2-1098240*C^8-\ 8600064*C^7*n-10644480*C^6*n^2-2280960*C^5*n^3-4435200*C^7-34594560*C^6*n-\ 48702720*C^5*n^2-15079680*C^4*n^3-485760*C^3*n^4-15996288*C^6-114652032*C^5*n-\ 154344960*C^4*n^2-44246400*C^3*n^3+1140480*C^2*n^4+419520*C*n^5-49452480*C^5-\ 301371840*C^4*n-329519520*C^3*n^2-46981440*C^2*n^3+17108160*C*n^4+1663200*n^5-\ 124853520*C^4-584526800*C^3*n-392055840*C^2*n^2+73053744*C*n^3+36590400*n^4-\ 240167400*C^3-720111480*C^2*n-43277992*C*n^2+199500840*n^3-311839484*C^2-\ 340717676*C*n+369066720*n^2-204495126*C+204495126*n, -45056*C^12-135168*C^11-\ 765952*C^10*n-675840*C^10-5406720*C^9*n-4967424*C^8*n^2-3294720*C^9-29956608*C^ 8*n-44605440*C^7*n^2-12367872*C^6*n^3-15206400*C^8-140912640*C^7*n-247256064*C^ 6*n^2-106444800*C^5*n^3-8321280*C^4*n^4-63985152*C^7-563126784*C^6*n-1000581120 *C^5*n^2-464851200*C^4*n^3-39536640*C^3*n^4+2085120*C^2*n^5-237371904*C^6-\ 1867574016*C^5*n-2991637440*C^4*n^2-1162529280*C^3*n^3-1877760*C^2*n^4+23696640 *C*n^5+665280*n^6-749121120*C^5-4925921440*C^4*n-6210420480*C^3*n^2-1245127104* C^2*n^3+401702400*C*n^4+61538400*n^5-1921339200*C^4-9602007360*C^3*n-7374402528 *C^2*n^2+1183799232*C*n^3+839306160*n^4-3742073808*C^3-11919298032*C^2*n-\ 1209981696*C*n^2+3716090400*n^3-4907883024*C^2-5773928160*C*n+6182918412*n^2-\ 3245265146*C+3245265146*n] The limits of the scaled central moments, starting at the third are: 1/2 2 2 1/2 (Pi - 4) 2 3 Pi - 4 Pi - 12 (7 Pi - 20 Pi - 16) 2 [-------------, -----------------, -------------------------, 3/2 2 5/2 (Pi - 2) (Pi - 2) (Pi - 2) 3 2 3 2 1/2 15 Pi - 6 Pi - 100 Pi - 40 (57 Pi - 126 Pi - 196 Pi - 48) 2 ----------------------------, -------------------------------------, 3 7/2 (Pi - 2) (Pi - 2) 4 3 2 105 Pi + 72 Pi - 952 Pi - 672 Pi - 112 -----------------------------------------, 4 (Pi - 2) 1/2 4 3 2 2 (561 Pi - 936 Pi - 2520 Pi - 1056 Pi - 128) ---------------------------------------------------, 9/2 (Pi - 2) 5 4 3 2 945 Pi + 1770 Pi - 10440 Pi - 11088 Pi - 3120 Pi - 288 ----------------------------------------------------------, 5 (Pi - 2) 5 4 3 2 1/2 (6555 Pi - 7590 Pi - 35640 Pi - 21648 Pi - 4400 Pi - 320) 2 ------------------------------------------------------------------, ( 11/2 (Pi - 2) 6 5 4 3 2 10395 Pi + 32580 Pi - 130020 Pi - 193248 Pi - 77616 Pi - 11968 Pi / 6 - 704) / (Pi - 2) ] / and in Maple notation [(Pi-4)*2^(1/2)/(Pi-2)^(3/2), (3*Pi^2-4*Pi-12)/(Pi-2)^2, (7*Pi^2-20*Pi-16)*2^(1 /2)/(Pi-2)^(5/2), (15*Pi^3-6*Pi^2-100*Pi-40)/(Pi-2)^3, (57*Pi^3-126*Pi^2-196*Pi -48)*2^(1/2)/(Pi-2)^(7/2), (105*Pi^4+72*Pi^3-952*Pi^2-672*Pi-112)/(Pi-2)^4, 2^( 1/2)*(561*Pi^4-936*Pi^3-2520*Pi^2-1056*Pi-128)/(Pi-2)^(9/2), (945*Pi^5+1770*Pi^ 4-10440*Pi^3-11088*Pi^2-3120*Pi-288)/(Pi-2)^5, (6555*Pi^5-7590*Pi^4-35640*Pi^3-\ 21648*Pi^2-4400*Pi-320)*2^(1/2)/(Pi-2)^(11/2), (10395*Pi^6+32580*Pi^5-130020*Pi ^4-193248*Pi^3-77616*Pi^2-11968*Pi-704)/(Pi-2)^6] and in Floats it is [-.9952717461, 3.869177308, -9.896966340, 34.76177935, -124.5213875, 495.397884\ 4, -2098.157939, 9476.097016, -45219.22019, 226972.5454] On the other hand The scaled central moments of the negative absolute value of the Normal dist\ ribution whose pdf is exp(-x^2/2)/sqrt(Pi/2)), supported in [-infinity..\ 0] are 1/2 2 2 1/2 (Pi - 4) 2 3 Pi - 4 Pi - 12 (7 Pi - 20 Pi - 16) 2 [-------------, -----------------, -------------------------, 3/2 2 5/2 (Pi - 2) (Pi - 2) (Pi - 2) 3 2 3 2 1/2 15 Pi - 6 Pi - 100 Pi - 40 (57 Pi - 126 Pi - 196 Pi - 48) 2 ----------------------------, -------------------------------------, 3 7/2 (Pi - 2) (Pi - 2) 4 3 2 105 Pi + 72 Pi - 952 Pi - 672 Pi - 112 -----------------------------------------, 4 (Pi - 2) 1/2 4 3 2 2 (561 Pi - 936 Pi - 2520 Pi - 1056 Pi - 128) ---------------------------------------------------, 9/2 (Pi - 2) 5 4 3 2 945 Pi + 1770 Pi - 10440 Pi - 11088 Pi - 3120 Pi - 288 ----------------------------------------------------------, 5 (Pi - 2) 5 4 3 2 1/2 (6555 Pi - 7590 Pi - 35640 Pi - 21648 Pi - 4400 Pi - 320) 2 ------------------------------------------------------------------, ( 11/2 (Pi - 2) 6 5 4 3 2 10395 Pi + 32580 Pi - 130020 Pi - 193248 Pi - 77616 Pi - 11968 Pi / 6 - 704) / (Pi - 2) ] / Yea! These are the same! It would be too complicated to actually list the explicit expressions for th\ e moments from, 13, to , 200, but let's check that The limits of the scaled central moments, up to the, 200 coincide with those of the negative absolute value of the Normal distributi\ on whose pdf is exp(-x^2/2)/sqrt(Pi/2)), supported in [-infinity..0] are Let's see: Too bad, they are not ----------------------------------- This ends this article that took, 32949.417, to produce