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.