Page 87 - Elements of Distribution Theory

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

3.2 Basic Properties 73

Example 3.5 (Uniform distribution on the unit interval). In Example 3.1 it is shown that
the characteristic function of this distribution is given by
1
exp(it) − 1

ϕ(t) = exp(itx) dx = .
0 it
Hence, part (ii) of Theorem 3.1 implies that for all −∞ < t < ∞,
| exp(it) − 1|≤|t|.

This is a useful result in complex analysis; see Appendix 2.

Example 3.6 (Normal distribution). Let Z denote a real-valued random variable with a
standard normal distribution and let µ and σ be real-valued constants with σ> 0. Define
a random variable X by

X = σ Z + µ.
The distribution of X is called a normal distribution with parameters µ and σ.
2
Recall that the characteristic function of Z is exp(−t /2); according to part (iii) of
Theorem 3.1, the characteristic function of X is
2
1 2 2 σ 2
exp(iµt)exp − σ t = exp − t + iµt .
2 2

Uniqueness and inversion of characteristic functions
The characteristic function is essentially the Fourier transform used in mathematical anal-
ysis. Let g denote a function of bounded variation such that
∞

|g(x)| dx < ∞.
−∞
The Fourier transform of g is given by
1 ∞
G(t) = √ g(x)exp{itx} dx, ∞ < t < ∞.
(2π)
−∞
The following result shows that it is possible to recover a function from its Fourier transform.
Theorem 3.2. Let G denote the Fourier transform of a function g, which is of bounded
variation.
(i) G(t) → 0 as t →±∞.
(ii) Suppose x 0 is a continuity point of g. Then
1 T
g(x 0 ) = √ lim G(t)exp{−itx 0 } dt.
(2π) T →∞ −T
Proof. The proof of this theorem uses the Riemann-Lebesgue Lemma (Section A3.4.10)
along with the result in Section A3.4.11 of Appendix 3.
Note that part (i) follows provided that

∞
lim g(x) sin(tx) dx = 0
t→∞
−∞