This for the set of 0-1 vectors of length n according to the number of 1's " The Average, Variance, and Scaled Moments Up to the, 6, -th of a Certain Combinatorial Random Variable By Shalosh B. Ekhad Theorem: Suppose that there is an infinite family of cominatorial families, \ on which there is a combinatorial random variable Let , P[n](w), be the weight-enumerator according to that random variable of\ the n-th member of our family. Suppose that you found out (e.g. by the transfer matrix method) that the bi-variate gener\ infinity ----- \ n 1 ating function, , ) P[n](w) t , equals , ------------- / 1 - (1 + w) t ----- n = 0 In other words infinity ----- \ n 1 ) P[n](w) t = ------------- / 1 - (1 + w) t ----- n = 0 Definition: Let a(n) be defined by the ordinary generating function infinity ----- \ n 1 ) a(n) t = ------- / 1 - 2 t ----- n = 0 then, of course, a(n), is the number of elements in the n-th family. The EXACT expression, in terms of a(n) and n, for the EXPECTATION is n/2 and in Maple notation 1/2*n The EXACT expression, in terms of a(n) and n, for the VARIANCE is n/4 and in Maple notation 1/4*n The EXACT expression, in terms of a(n) and n, for the, 3, -th moment about the mean is 0 and in Maple notation 0 The EXACT expression, in terms of a(n) and n, for the, 4, -th moment about the mean is n (3 n - 2) ----------- 16 and in Maple notation 1/16*n*(3*n-2) The EXACT expression, in terms of a(n) and n, for the, 5, -th moment about the mean is 0 and in Maple notation 0 The EXACT expression, in terms of a(n) and n, for the, 6, -th moment about the mean is 2 n (15 n - 30 n + 16) --------------------- 64 and in Maple notation 1/64*n*(15*n^2-30*n+16) Let , b, be the largest positive root of the polynomial equation -b + 2 = 0 and in Maple notation -b+2 = 0 whose floating-point approximation is 2. Then the size of the n-th family (i.e. straight enumeration) is very close t\ o n b and in Maple notation b^n and in floating point n 2. The average of the statistics is, asymptotically n/2 and in Maple notation 1/2*n and in floating-point 1/2*n The variance of the statistics is, asymptotically n/4 and in Maple notation 1/4*n and in floating-point 1/4*n The skewness of the statistics is, asymptotically 0 and in Maple notation 0 and in floating-point 0 The kurtosis of the statistics is, asymptotically 3 - 2/n and in Maple notation 3-2/n and in floating-point 3-2/n The standardized, 5, -th moment (about the mean) of the statistics is, asymptotically 0 and in Maple notation 0 and in floating-point 0 The standardized, 6, -th moment (about the mean) of the statistics is, asymptotically 30 16 15 - ---- + ---- n 2 n and in Maple notation 15-30/n+16/n^2 and in floating-point 15-30/n+16/n^2 Finally here is the asymptotic expressions, to order 2, of the standarized \ third to, 6, -th moment The , 3, -th standardized moment is, 0 and in floating-point, 0 The , 4, -th standardized moment is, 3 - 2/n and in floating-point, 3 - 2/n The , 5, -th standardized moment is, 0 and in floating-point, 0 30 16 The , 6, -th standardized moment is, 15 - ---- + ---- n 2 n 30 16 and in floating-point, 15 - ---- + ---- n 2 n This took, 0.075, seconds.