Page 137 - Elements of Distribution Theory

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052184472Xc04 CUNY148/Severini May 24, 2005 2:39

4.5 Moments and Cumulants of the Sample Mean 123

It follows from the results in Example 4.25 that
2r
E X = (2r)!, r = 1, 2,....
1
Hence, the first four moments of T n are given by E(T n ) = 2,
20 120 592
2 3
E T = 4 + , E T = 8 + +
n n 2
n n n
and
480 5936 31584
4
E T = 16 + + + .
n 2 3
n n n
¯
Central moments of X n
Results analogous to those given in Theorem 4.17 for the moments can be obtained for
3
4
2
the central moments by taking E(X 1 ) = 0 and then interpreting E(X ), E(X ), and E(X )
1 1 1
in Theorem 4.17 as central moments. The resulting expressions are given in the following
corollary; the proof is left as an exercise.
Corollary 4.3. Let X 1 , X 2 ,..., X n denote independent, identically distributed, real-valued
random variables and let
n
1
¯
X n = X j .
n
j=1
4
Assume that E[X ] < ∞. Let µ = E(X 1 ), and let µ 2 ,µ 3 ,µ 4 denote the second, third, and
1
¯
¯
¯
fourth central moments, respectively, of X 1 . Let µ 2 (X n ),µ 3 (X n ), and µ 4 (X n ) denote the
¯
¯
second, third, and fourth central moments, respectively, of X n . Then E(X n ) = µ,
1 1
¯ ¯
µ 2 (X n ) = µ 2 , µ 3 (X n ) = µ 3
n n 2
and
3(n − 1) 1 3 1
¯ 2 2 2
µ 4 (X n ) = µ + µ 4 = µ + µ 4 − 3µ .
2
2
2
n 3 n 3 n 2 n 3
Example 4.27 (Standard exponential distribution). As in Examples 4.25 and 4.26, let
X 1 , X 2 ,..., X n denote independent, identically distributed, standard exponential random
variables. It is straightforward to show that the first four central moments of the standard
exponential distribution are 0, 1, 2, 9; these may be obtained using the expressions for
central moments in terms of cumulants, given in Section 4.4. It follows from Corollary 4.3
¯
2
that the first four central moments of X n are 0, 1/n,2/n , and
3 6
+ ,
n 2 n 3
respectively.
¯ k
We can see from Corollary 4.3 that, as k increases, the order of E[(X n − µ) ]as n →∞
is a nondecreasing power of 1/n:
1
1 1
¯
¯
¯
4
3
2
E[(X n − µ) ] = O , E[(X n − µ) ] = O , E[(X n − µ) ] = O .
n n 2 n 2
The following theorem gives a generalization of these results.