Page 91 - Elements of Distribution Theory

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052184472Xc03 CUNY148/Severini May 24, 2005 2:34

3.2 Basic Properties 77

Example 3.7 (Chi-squared distribution). Let Z denote a random variable with a standard
2
normal distribution and consider the distribution of Z . This distribution has characteristic
function
∞ 1 1 1 ∞ 1

2 2 2
exp(itz )√ exp − z dz = √ exp − (1 − 2it)z dz
(2π) 2 (2π) 2
−∞ −∞
1
= .
1
(1 − 2it) 2
Now consider independent standard normal random variables Z 1 , Z 2 , ··· , Z n and let
2
2
X = Z +· · · + Z .By Theorem 3.4, the characteristic function of X is
1
n
1
ϕ(t) = n .
(1 − 2it) 2
Comparing this result to the characteristic function derived in Example 3.4 shows that X
has a gamma distribution with parameters α = n/2 and β = 1/2. This special case of the
gamma distribution is called the chi-squared distribution with n degrees of freedom; note
that this distribution is defined for any positive value of n, not just integer values.
Example 3.8 (Sum of binomial random variables). Let X 1 and X 2 denote independent
random variables such that, for j = 1, 2, X j has a binomial distribution with parameters n j
and θ j . Recall from Example 3.3 that X j has characteristic function

n j
ϕ j (t) = [1 − θ j + θ j exp(it)] , j = 1, 2.
Let X = X 1 + X 2 . Then X has characteristic function

ϕ(t) = [1 − θ 1 + θ 1 exp(it)] [1 − θ 2 + θ 2 exp(it)] .
n 1
n 2
Hence, if θ 1 = θ 2 , then X also has a binomial distribution.
An expansion for characteristic functions
It is well known that the exponential function of a real-valued argument, exp(x), can be
expanded in a power series in x:
∞ j
x
exp(x) = .
j!
j=0
The same result holds for complex arguments, so that
∞ j
(itx)
exp(itx) = ;
j!
j=0
see Appendix 2 for further discussion. Thus, the characteristic function of a random variable
X can be expanded in power series whose coefficients involve expected values of the form
m
E(X ), m = 0, 1,....
m
This fact can be used to show that the existence of E(X ), m = 1, 2,..., is related to
m
the smoothness of the characteristic function at 0; in particular, if E(|X| ) < ∞, then ϕ X
is m-times differentiable at 0. The converse to this result is also useful, but it applies only
to moments, and derivatives, of even order. Specifically, if ϕ X is 2m-times differentiable at
0 then E(X 2m ) < ∞. The details are given in the following theorem.