Page 105 - Elements of Distribution Theory

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3.4 Exercises 91

3.2 Let X denote a real-valued random variable with an absolutely continuous distribution with
density function
1
p(x) = exp(−|x|), −∞ < x < ∞.
2
This is the standard Laplace distribution. Find the characteristic function of X.
3.3 Let X denote a real-valued random variable with a uniform distribution on the interval (a, b),
b > a. That is, X has an absolutely continuous distribution with density function
1
p(x) = , a < x < b;
b − a
here a and b are fixed constants. Find the characteristic function of X.
3.4 Let X denote a real-valued random variable with distribution function F and characteristic
function ϕ. Suppose that ϕ satisfies the following condition:
1 T
lim ϕ(t) dt = 2.
T →∞ T
−T
Based on this, what can you conclude about the distribution of X.Beas specific as possible.
3.5 Let X 1 and X 2 denote independent real-valued random variables with distribution functions F 1 ,
F 2 , and characteristic functions ϕ 1 , ϕ 2 , respectively. Let Y denote a random variable such that
X 1 , X 2 , and Y are independent and

Pr(Y = 0) = 1 − Pr(Y = 1) = α
for some 0 <α < 1. Define

X 1 if Y = 0
Z = .
X 2 if Y = 1
Find the characteristic function of Z in terms of ϕ 1 ,ϕ 2 , and α.
3.6 Let X denote a real-valued random variable with characteristic function ϕ. Suppose that
|ϕ(1)|= ϕ(π) = 1.

Describe the distribution of X;beas specific as possible.
3.7 Prove Theorem 3.4.
3.8 Let X 1 and X 2 denote independent random variables each with a standard normal distribution
and let Y = X 1 X 2 . Find the characteristic function and density function of Y.
3.9 Prove Theorem 3.6.
3.10 Let X 1 and X 2 denote independent random variables, each with the standard Laplace distribution;
see Exercise 3.2. Let Y = X 1 + X 2 . Find the characteristic function and density function of Y.
3.11 Prove Corollary 3.2.
3.12 Let X denote a real-valued random variable with characteristic function ϕ. Suppose that g is a
real-valued function on R that has the representation

∞
g(x) = G(t)exp(itx) dt
−∞
for some function G satisfying

∞
|G(t)| dt < ∞.
−∞