Page 142 - Elements of Distribution Theory

P1: JZP
052184472Xc04 CUNY148/Severini May 24, 2005 2:39

128 Moments and Cumulants

4.7 Let X denote a d-dimensional random vector with covariance matrix satisfying | | < ∞.
Show that X has a nondegenerate distribution if and only if is positive definite.
4.8 Let X and Y denote real-valued random variables such that X has mean µ X and standard
deviation σ X , Y has mean µ Y and standard deviation σ Y , and X and Y have correlation ρ.
(a) Find the value of β ∈ R that minimizes Var(Y − βX).
(b) Find the values of β ∈ R such that Y and Y − βX are uncorrelated.
(c) Find the values of β ∈ R such that X and Y − βX are uncorrelated.
(d) Find conditions under which, for some β, Y − βX is uncorrelated with both X and Y.
(e) Suppose that E(Y|X) = α + βX for some constants α, β. Express α and β in terms of
µ X ,µ Y ,σ X ,σ Y ,ρ.
2
2
4.9 Let X and Y denote real-valued random variables such that E(X ) < ∞ and E(Y ) < ∞. Sup-
pose that E[X|Y] = 0. Does it follow that ρ(X, Y) = 0?
2
4.10 Let X and Y denote real-valued identically distributed random variables such that E(X ) < ∞.
Give conditions under which
2
2
ρ(X, X + Y) ≥ ρ(X, Y) .
2
2
4.11 Let X and Y denote real-valued random variables such that E(X ) < ∞ and E(Y ) < ∞ and
let ρ denote the correlation of X and Y. Find the values of ρ for which
2
E[(X − Y) ] ≥ Var(X).
4.12 Let X denote a nonnegative, real-valued random variable; let F denote the distribution function
of X and let L denote the Laplace transform of X. Show that

∞
L(t) = t exp(−tx)F(x) dx, t ≥ 0.
0
4.13 Prove Theorem 4.7.
4.14 Let X denote a nonnegative, real-valued random variable and let L(t) denote the Laplace trans-
form of X. Show that
d n
(−1) n L(t) ≥ 0, t ≥ 0.
dt n
A function with this property is said to be completely monotone.
4.15 Let X denote a random variable with frequency function
x
p(x) = θ(1 − θ) , x = 0, 1, 2,...
where 0 <θ < 1.
Find the moment-generating function of X and the first three moments.
r
4.16 Let X denote a real-valued random variable and suppose that, for some r = 1, 2,..., E(|X| ) =
m
∞. Does it follow that E(|X| ) =∞ for all m = r + 1,r + 2,...?
4.17 Prove Theorem 4.10.
4.18 Prove Theorem 4.11.
4.19 Let X denote a random variable with the distribution given in Exercise 4.15. Find the cumulant-
generating function of X and the first three cumulants.
4.20 Let X denote a random variable with a gamma distribution, as in Exercise 4.1. Find the cumulant-
generating function of X and the first three cumulants.
4.21 Let Y be a real-valued random variable with distribution function F and moment-generating
function M(t), |t| <δ, where δ> 0is chosen to be as large as possible. Define
β = inf{M(t): 0 ≤ t <δ}.