Generalization of the Central Limit Theorem for Repeated Throws of a fair k-faced die, for k between 2(coin) and , 6 by Shalosh B. Ekhad Theorem Number , 2, : Consider the r.v. total number of dots obtained on rolling n times a fair die, with, 2, faces such that the faces have 1, 2, ..., , 2, dots The even, 2r-th, moment, about the mean is (asymptotically, ignoring 1 ----, and beyond): 4 n / r | (r - 1) r r (r - 1) (r - 2) (5 r + 1) (n/4) (2 r)! |1 - --------- + --------------------------- | 3 n 2 \ 90 n 2 \ (r - 1) (r - 2) (r - 3) (35 r + 21 r - 32) r| / r - ---------------------------------------------| / (r! 2 ) 3 | / 5670 n / The variance is, n/4 After dividing by the variance to the power r (i.e. normalizing it) we get that the normalized 2r-th moment is / | (r - 1) r r (r - 1) (r - 2) (5 r + 1) (2 r)! |1 - --------- + --------------------------- | 3 n 2 \ 90 n 2 \ (r - 1) (r - 2) (r - 3) (35 r + 21 r - 32) r| / r - ---------------------------------------------| / (r! 2 ) 3 | / 5670 n / In particular, it follows that this discrete prob. dist. is asymptotically normal, but it does much more! We have established the asymptotic to, 4, terms . Of course, by symmetry, all the odd moments are zero but we can derive this directly, getting that the odd moments eqaul 0 Theorem Number , 3, : Consider the r.v. total number of dots obtained on rolling n times a fair die, with, 3, faces such that the faces have 1, 2, ..., , 3, dots The even, 2r-th, moment, about the mean is (asymptotically, ignoring 1 ----, and beyond): 4 n / /2 n\r | (r - 1) r r (r - 1) (r - 2) (15 r + 7) |---| (2 r)! |1 - --------- + ---------------------------- \ 3 / | 4 n 2 \ 480 n 2 \ (r - 1) (r - 2) (r - 3) (35 r + 49 r - 18) r| / r - ---------------------------------------------| / (r! 2 ) 3 | / 13440 n / 2 n The variance is, --- 3 After dividing by the variance to the power r (i.e. normalizing it) we get that the normalized 2r-th moment is / | (r - 1) r r (r - 1) (r - 2) (15 r + 7) (2 r)! |1 - --------- + ---------------------------- | 4 n 2 \ 480 n 2 \ (r - 1) (r - 2) (r - 3) (35 r + 49 r - 18) r| / r - ---------------------------------------------| / (r! 2 ) 3 | / 13440 n / In particular, it follows that this discrete prob. dist. is asymptotically normal, but it does much more! We have established the asymptotic to, 4, terms . Of course, by symmetry, all the odd moments are zero but we can derive this directly, getting that the odd moments eqaul 0 Theorem Number , 4, : Consider the r.v. total number of dots obtained on rolling n times a fair die, with, 4, faces such that the faces have 1, 2, ..., , 4, dots The even, 2r-th, moment, about the mean is (asymptotically, ignoring 1 ----, and beyond): 4 n / /5 n\r | 17 (r - 1) r r (r - 1) (r - 2) (289 r + 173) |---| (2 r)! |1 - ------------ + ------------------------------- \ 4 / | 75 n 2 \ 11250 n 2 \ 17 (r - 1) (r - 2) (r - 3) (2023 r + 3633 r - 640) r| / r - -----------------------------------------------------| / (r! 2 ) 3 | / 17718750 n / 5 n The variance is, --- 4 After dividing by the variance to the power r (i.e. normalizing it) we get that the normalized 2r-th moment is / | 17 (r - 1) r r (r - 1) (r - 2) (289 r + 173) (2 r)! |1 - ------------ + ------------------------------- | 75 n 2 \ 11250 n 2 \ 17 (r - 1) (r - 2) (r - 3) (2023 r + 3633 r - 640) r| / r - -----------------------------------------------------| / (r! 2 ) 3 | / 17718750 n / In particular, it follows that this discrete prob. dist. is asymptotically normal, but it does much more! We have established the asymptotic to, 4, terms . Of course, by symmetry, all the odd moments are zero but we can derive this directly, getting that the odd moments eqaul 0 Theorem Number , 5, : Consider the r.v. total number of dots obtained on rolling n times a fair die, with, 5, faces such that the faces have 1, 2, ..., , 5, dots The even, 2r-th, moment, about the mean is (asymptotically, ignoring 1 ----, and beyond): 4 n / r | 13 (r - 1) r r (r - 1) (r - 2) (169 r + 113) (2 n) (2 r)! |1 - ------------ + ------------------------------- | 60 n 2 \ 7200 n 2 \ 13 (r - 1) (r - 2) (r - 3) (1183 r + 2373 r - 250) r| / r - -----------------------------------------------------| / (r! 2 ) 3 | / 9072000 n / The variance is, 2 n After dividing by the variance to the power r (i.e. normalizing it) we get that the normalized 2r-th moment is / | 13 (r - 1) r r (r - 1) (r - 2) (169 r + 113) (2 r)! |1 - ------------ + ------------------------------- | 60 n 2 \ 7200 n 2 \ 13 (r - 1) (r - 2) (r - 3) (1183 r + 2373 r - 250) r| / r - -----------------------------------------------------| / (r! 2 ) 3 | / 9072000 n / In particular, it follows that this discrete prob. dist. is asymptotically normal, but it does much more! We have established the asymptotic to, 4, terms . Of course, by symmetry, all the odd moments are zero but we can derive this directly, getting that the odd moments eqaul 0 Theorem Number , 6, : Consider the r.v. total number of dots obtained on rolling n times a fair die, with, 6, faces such that the faces have 1, 2, ..., , 6, dots The even, 2r-th, moment, about the mean is (asymptotically, ignoring 1 ----, and beyond): 4 n / /35 n\r | 37 (r - 1) r r (r - 1) (r - 2) (28749 r + 20393) |----| (2 r)! |1 - ------------ + ----------------------------------- \ 12 / | 175 n 2 \ 1286250 n 2 \ 37 (r - 1) (r - 2) (r - 3) (9583 r + 20393 r - 1440) r| / r - -------------------------------------------------------| / (r! 2 ) 3 | / 225093750 n / 35 n The variance is, ---- 12 After dividing by the variance to the power r (i.e. normalizing it) we get that the normalized 2r-th moment is / | 37 (r - 1) r r (r - 1) (r - 2) (28749 r + 20393) (2 r)! |1 - ------------ + ----------------------------------- | 175 n 2 \ 1286250 n 2 \ 37 (r - 1) (r - 2) (r - 3) (9583 r + 20393 r - 1440) r| / r - -------------------------------------------------------| / (r! 2 ) 3 | / 225093750 n / In particular, it follows that this discrete prob. dist. is asymptotically normal, but it does much more! We have established the asymptotic to, 4, terms . Of course, by symmetry, all the odd moments are zero but we can derive this directly, getting that the odd moments eqaul 0 This took, 6.940, seconds