Page 143 - Elements of Distribution Theory

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

4.7 Exercises 129

Suppose that there exists a unique real number τ ∈ (0,δ) such that M(τ) = β.
(a) Show that Pr(Y ≥ 0) ≤ β.
(b) Show that M (τ) = 0.

(c) Let
1
x
G(x) = exp(τy) dF(y), −∞ < x < ∞.
β −∞
Note that G is a distribution function on R and let X denote a random variable with
distribution function G. Find the moment-generating function of X.
(d) Find E(X).
4.22 Let X denote a real-valued random variable with distribution function F. Let K(t), |t| <δ,
δ> 0, denote the cumulant-generating function of X; assume that δ is chosen to be as large as
possible.
Let a ≥ 0beafixed, real-valued constant and define a function
K a (t) = K(t) − at, t ≥ 0.

Define
ρ a = inf{K a (t): 0 ≤ t <δ}.

(a) Calculate ρ a ,asa function of a, for the standard normal distribution and for the Poisson
distribution with mean 1.
(b) Show that
Pr(X ≥ a) ≤ exp(ρ a ).

(c) Let X 1 , X 2 ,..., X n denote independent random variables, each with the same distribution
as X. Obtain a bound for

X 1 + ··· + X n
Pr ≥ a
n
that generalizes the result given in part (b).
4.23 Let X denote a real-valued random variable with moment-generating function M X (t), |t| <δ,
δ> 0. Suppose that the distribution of X is symmetric about 0; that is, suppose that X and −X
have the same distribution. Find κ j (X), j = 1, 3, 5,....
4.24 Consider a distribution on the real line with moment-generating function M(t), |t| <δ, δ> 0
r
and cumulants κ 1 ,κ 2 ,.... Suppose that E(X ) = 0, r = 1, 3, 5,.... Show that
κ 1 = κ 3 = ··· = 0.
Does the converse hold? That is, suppose that all cumulants of odd order are 0. Does it follow
that all moments of odd order are 0?
4.25 Let X and Y denote real-valued random variables and assume that the moment-generating
function of (X, Y)exists. Write
1
2 2
M(t 1 , t 2 ) = E[exp(t 1 X + t 2 Y)], t + t 2 2 <δ,
1
K(t 1 , t 2 ) = log M(t 1 , t 2 ), and let κ ij , i, j = 0, 1,... , denote the joint cumulants of (X, Y).
Let S = X 1 + X 2 and let K S denote the cumulant-generating function of S.
(a) Show that
√
K S (t) = K(t, t), |t|≤ δ/ 2.