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3.3 Further Properties of Characteristic Functions 87

Example 3.14 (Uniform distribution on the unit interval). Consider the uniform distri-
bution on the unit interval; the characteristic function of this distribution is given by
1
exp(it) − 1 sin(t) cos(t) − 1
ϕ(t) = exp(itx) dx = = − i , −∞ < t < ∞.
0 it t t
Hence,
2
sin(t) + (cos(t) − 1) 2
2 −2
|ϕ(t)| = = O(|t| )as |t|→∞
t 2
so that the density of this distribution is differentiable, at most, one time. Here p(x) = 1if
0 < x < 1 and p(x) = 0 otherwise so that p is not differentiable.

Example 3.15 (Binomial distribution). Consider a binomial distribution with parameters
n and θ. The characteristic function of this distribution is given by

ϕ(t) = [1 − θ + θ exp(it)] n
so that
n
2
2
|ϕ(t)|= [(1 − θ) + 2θ(1 − θ) cos(t) + θ ] 2 .
It is easy to see that
n
lim inf |ϕ(t)|=|2θ − 1| and lim sup |ϕ(t)|= 1
|t|→∞
|t|→∞
so that ϕ(t) does not have a limit as |t|→ 0; see Figure 3.2 for a graphical illustration of
this fact. It follows that the binomial distribution is not absolutely continuous, which, of
course, is obvious from its definition.

Symmetric distributions
The distribution of X is said to be symmetric about a point x 0 if X − x 0 and −(X − x 0 )
have the same distribution. Note that this implies that

F(x 0 + x) = 1 − F((x 0 − x)−), −∞ < x < ∞,
where F denotes the distribution function of X.
The following theorem shows that the imaginary part of the characteristic function of a
distribution symmetric about 0 is 0; the proof is left as an exercise.

Theorem 3.10. Let X denote a real-valued random variable with characteristic fun-
ction ϕ. The distribution of X is symmetric about 0 if and only if ϕ(t) is real for all
−∞ < t < ∞.

Example 3.16 (Symmetrization of a distribution). Let X denote a real-valued random
variable with characteristic function ϕ. Let X 1 , X 2 denote independent random variables,
such that X 1 and X 2 each have the same distribution as X, and let Y = X 1 − X 2 . Then Y
has characteristic function
ϕ Y (t) = E[exp(it X 1 )exp(−it X 2 )] = ϕ(t)ϕ(−t)
.
2
=|ϕ(t)| , −∞ < t < ∞