Page 85 - Elements of Distribution Theory

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

3.1 Introduction 71

Thus, aside from a constant, the characteristic function of the standard normal distribution
is the same as its density function.

Example 3.3 (Binomial distribution). Let X denote a real-valued random variable with a
discrete distribution with frequency function

n x n−x
p(x) = θ (1 − θ) , x = 0,..., n;
x
here θ and n are fixed constants, with θ taking values in the interval (0, 1) and n taking
values in the set {1, 2,...}. This is a binomial distribution with parameters n and θ. The
characteristic function of this distribution is given by

n
n x n−x
ϕ(t) = exp(itx) θ (1 − θ)
x
x=0
n θ exp(itx) x
n

= (1 − θ) n
x 1 − θ
x=0
n
= [1 − θ + θ exp(it)] .
Plots of the real and imaginary parts of ϕ for the case n = 3,θ = 1/2 are given in
Figure 3.2.

Real Part

u (t)

−
− − −
t
Imaginary Part

v (t)

−
− − −
t
Figure 3.2. Characteristic function in Example 3.3.