Page 144 - Elements of Distribution Theory

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130 Moments and Cumulants

(b) Let κ j (S), j = 1, 2,..., denote the cumulants of S. Show that
κ 2 (S) = κ 20 + 2κ 11 + κ 02 .
(c) Derive a general expression for κ j (S)in terms of κ ik , i, k = 1, 2,....
4.26 Let X and Y denote discrete random variables, with ranges X and Y, respectively. Suppose that
X and Y each contain m distinct elements of R, for some m = 1, 2,... ; assume that, for each
x ∈ X and each y ∈ Y,
Pr(X = x) > 0 and Pr(Y = y) > 0.
Suppose that
j
j
E(X ) = E(Y ), j = 1, 2,..., m.
Does it follow that X and Y have the same distribution? Why or why not?
4.27 Let (X, Y) denote a two-dimensional random vector with joint cumulants κ ij , i, j = 1, 2,....
Given an expression for Var(XY)in terms of the κ ij .
4.28 Let X 1 , X 2 ,... denote a sequence of real-valued random variables such that X 1 , X 2 ,... are
exchangeable and (X 1 , X 2 ,...)isa martingale. Find the correlation of X i and X j , i, j =
1, 2,....
4.29 For each n = 1, 2,..., let (X 1 , Y 1 ), (X 2 , Y 2 ),..., (X n , Y n ) denote independent, identically
distributed, two-dimensional random vectors with joint cumulants κ ij , i, j = 1, 2,.... Let
n n −r
¯ X = X j /n and ¯ Y = Y j /n. Find r = 1, 2,... such that Var( ¯ X ¯ Y) = O(n )as
j=1 j=1
n →∞.
4.30 Let (X 1 , Y 1 ),..., (X n , Y n ) denote independent, identically distributed random vectors such that,
for each j, X j and Y j are real-valued; assume that all moments of (X 1 , Y 1 )exist and are finite.
Let
1 n 1 n
¯ X = X j and ¯ Y = Y j .
n n
j=1 j=1
2
(a) Express E( ¯ X ¯ Y) and E( ¯ X ¯ Y)in terms of the moments of (X 1 , Y 1 ).
(b) Express the cumulants of ( ¯ X, ¯ Y)of orders (1, 1) and (2, 1) in terms of the cumulants of
(X 1 , Y 1 ).
4.31 Let X and Y denote real-valued random variables. Let κ 1 (Y),κ 2 (Y),... denote the cumulants of
the conditional distribution of X given Y and let κ 1 ,κ 2 ,... denote the cumulants of the marginal
distribution of X.
(a) Show that E[κ 1 (Y)] ≤ κ 1 and E[κ 2 (Y)] ≤ κ 2 .
(b) Does the same result hold for κ 3 (Y) and κ 3 ? That is, is it true that E[κ 3 (Y)] ≤ κ 3 ?
2
2
2
4.32 Let X, Y, and Z denote real-valued random variables such that E(X ), E(Y ), and E(Z ) are all
finite. Find an expression for Cov(X, Y)in terms of Cov(X, Y|Z), E(X|Z), and E(Y|Z).
2
4.33 Let X 1 ,..., X n denote real-valued, exchangeable random variables such that E(X ) < ∞. Let
1

S = X j .For 1 ≤ i < j ≤ n, find the conditional correlation of X i and X j given S.
4.8 Suggestions for Further Reading
Moments and central moments, particularly the mean and variance, are discussed in nearly every
book on probability. Laplace transforms are considered in detail in Feller (1971, Chapters XIII and
XIV); see also Billingsley (1995, Section 22) and Port (1994, Chapter 50). Laplace transforms are
often used in nonprobabilistic contexts; see, for example, Apostol (1974, Chapter 11) and Widder
(1971). Moment-generating functions are discussed in Port (1994, Chapter 56); see Lukacs (1960)