Page 106 - Elements of Distribution Theory

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92 Characteristic Functions

Show that
∞

E[g(X)] = G(t)ϕ(t) dt.
−∞
3.13 Consider a distribution on the real line with characteristic function
2
2
ϕ(t) = (1 − t /2) exp(−t /4), −∞ < t < ∞.
Show that this distribution is absolutely continuous and find the density function of the distri-
bution.
3.14 Let ϕ 1 ,...,ϕ n denote characteristic functions for distributions on the real line. Let a 1 ,..., a n
denote nonnegative constants such that a 1 + ··· + a n = 1. Show that

ϕ(t) = n a j ϕ j (t), −∞ < t < ∞
j=1
is also a characteristic function.
3.15 Let X and Y denote independent, real-valued random variables each with the same marginal
distribution and let ϕ denote the characteristic function of that distribution. Consider the random
vector (X + Y, X − Y) and let ϕ 0 denote its characteristic function. Show that
2
ϕ 0 ((t 1 , t 2 )) = ϕ(t 1 + t 2 )ϕ(t 1 − t 2 ), (t 1 , t 2 ) ∈ R .
3.16 Consider an absolutely continuous distribution on the real line with density function p. Suppose
that p is piece-wise continuous with a jump at x 0 , −∞ < x 0 < ∞. Show that
∞

|ϕ(t)| dt =∞,
−∞
where ϕ denotes the characteristic function of the distribution.
3.17 Suppose that X is a real-valued random variable. Suppose that there exists a constant M > 0
such that the support of X lies entirely in the interval [−M, M]. Let ϕ denote the characteristic
function of X. Show that ϕ is infinitely differentiable at 0.
3.18 Prove Corollary 3.3.
3.19 Let X denote a real-valued random variable with characteristic function
1
ϕ(t) = [cos(t) + cos(tπ)], −∞ < t < ∞.
2
(a) Is the distribution of X absolutely continuous?
r
(b) Does there exist an r such that E(X ) either does not exist or is infinite?
3.20 Let ϕ denote the characteristic function of the distribution with distribution function

0 if x < 0
F(x) = .
1 − exp(−x), if 0 ≤ x < ∞
Show that this distribution is absolutely continuous and that

∞
|ϕ(t)| dt =∞.
−∞
Thus, the converse to Theorem 3.8 does not hold.
3.21 Find the density function of the distribution with characteristic function

1 −|t| if |t|≤ 1
ϕ(t) = .
0 otherwise