Theory of Statistics - George Mason University

2.11 The Family of Normal Distributions 191
theorem. This form of the theorem and various preliminary forms leading up
to it are proved beginning on page 426.
From the family of central chi-squared distributions together with an independent
normal family, we get the family of t distributions (central or noncentral,
depending on the mean of the normal). From the family of chi-squared
distributions (central or noncentral) we get the family of F distributions (central,
or singly or doubly noncentral; see Example 1.15 on page 59 for the
central distributions).
The expectations of reciprocals of normal random variables have interesting
properties. First of all, we see that for X ∼ N(0, 1), E(1/X) does not exist.
Now, for X ∼ Nd(0, I) consider
1
E
X2
. (2.40)
2
For d ≤ 2, this expectation is infinite (exercise). For d ≥ 3, however, this
expectation is finite (exercise).
2.11.3 Characterizations of the Normal Family of Distributions
A simple characterization of a normal distribution was proven by Cramér in
1936:
Theorem 2.4
Let X1 and X2 be independent random variables. Then X1 and X2 have normal
distributions if and only if their sum X1 + X2 has a normal distribution.
Proof. ***fix
The independence of certain functions of random variables imply that
those random variables have a normal distribution; that is, the independence
of certain functions of random variables characterize the normal distribution.
Theorem 2.5 (Bernstein’s theorem)
Let X1 and X2 be iid random variables with nondegenerate distributions, and
let Y1 = X1 + X2 and Y2 = X1 − X2. If Y1 and Y2 are also independent then
X1 and X2 have normal distributions.
Proof. ***fix
An extension of Bernstein’s theorem is the Darmois theorem, also called
the Darmois-Skitovich theorem.
Theorem 2.6 (Darmois theorem)
Let X1, . . ., Xn be iid random variables with nondegenerate distributions, and
let
n
Y1 =
and
i=1
biXi
Theory of Statistics c○2000–2013 James E. Gentle