Theory of Statistics - George Mason University

60 1 Probability Theory
Example 1.16 distribution of the sum of squares of independent
standard normal random variables
iid 2 Suppose X1, . . ., Xn ∼ N(0, 1), and let Y = Xi . In Example 1.13, we saw
that Yi = X2 d
i = χ2 1. Because the Xi are iid, the Yi are iid. Now the MGF of
a χ2 1 is
E e tYi
∞
1
= √ y
0 2π −1/2 e −y(1−2t)/2 dy
= (1 − 2t) −1/2
for t < 1
2 .
Hence, the MGF of Y is (1 − 2t) n/2 for t < 1/2, which is seen to be the MGF
of a chi-squared random variable with n degrees of freedom.
This is a very important result for applications in statistics.
1.1.11 Decomposition of Random Variables
We are often interested in the sum of random numbers,
Sk = X1 + · · · + Xk. (1.131)
Because the sum may grow unwieldy as k increases, we may work with normed
sums of the form Sk/ak.
In order to develop interesting properties of Sk, there must be some commonality
among the individual Xi. The most restrictive condition is that the
Xi be iid. Another condition for which we can develop meaningful results is
that the Xi be independent, but different subsequences of them may have
different distributions.
The finite sums that we consider in this section have relevance in the limit
theorems discussed in Sections 1.4.1 and 1.4.2.
Infinitely Divisible Distributions
Instead of beginning with the Xi and forming their sum, we can think of
the problem as beginning with a random variable X and decomposing it into
additive components. A property of a random variable that allows a particular
kind of additive decomposition is called divisibility.
Definition 1.31 (n-divisibility of random variables)
Given a random variable X and an integer n ≥ 2, we say X is n-divisible if
there exist iid random variables X1, . . ., Xn such that
X d = X1 + · · · + Xn.
Theory of Statistics c○2000–2013 James E. Gentle