Page 51 - Elements of Distribution Theory

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1.9 Exercises 37

1.23 Prove the Cauchy-Schwarz inequality.
2
Hint: E{[g 1 (X) − tg 2 (X)] }≥ 0 for all t. Find the minimum of this expression.
1.24 Prove Jensen’s inequality.
Hint: For a convex function g, show that there exists a constant c such that
g(x) ≥ g(E(X)) + c(x − E(X))
for all x ∈ R.
1.25 Prove Markov’s inequality.
1.26 Let X denote a real-valued random variable with distribution function F; assume that E(|X|) <
∞. Show that
0
∞
E(X) = (1 − F(x)) dx − F(x) dx
0 −∞

∞
= [1 − F(x) − F(−x)] dx.
0
1.27 Let X denote a real-valued random variable with distribution function F. Find the distribution
functions of |X| and X , where
+
X if X > 0

+
X = .
0 otherwise
2
1.28 Let L 2 denote the set of real-valued random variables X satisfying E(X ) < ∞. Show that
L 2 is a linear space: if X 1 and X 2 are elements of L 2 and a and b are scalar constants, then
aX 1 + bX 2 ∈ L 2 .
1.29 Consider the space of random variables L 2 described in Exercise 1.28. Let 0 denote the random
variable identically equal to 0 and, for X ∈ L 2 , write X = 0ifPr(X = 0) = 1. Define a function
2
2
||·|| on L 2 as follows: for X ∈ L 2 , ||X|| = E(X ). Show that ||·|| defines a norm on L 2 : for
all X 1 and X 2 in L 2 and all scalar constants a,
(a) ||X 1 || ≥ 0 and ||X 1 || = 0if and only if X 1 = 0
(b) ||X 1 + X 2 ||≤||X 1 ||+||X 2 ||
(c) ||aX||=|a|||X||
1.30 Let X denote a real-valued random variable and suppose that the distribution of X is symmetric
about 0; that is, suppose that X and −X have the same distribution. Show that, for r = 1, 3,...,
r
r
E(X ) = 0 provided that E(X )exists.
1.31 Let X be real-valued random variable with a discrete distribution with frequency function
x
p(x) = λ exp(−λ)/x!, x = 0, 1, 2,...
where λ> 0; this is a Poisson distribution with parameter λ. Find E(X).
1.32 Let X denote a real-valued random variable with an absolutely continuous distribution with
r
density αx α−1 ,0 < x < 1. Find E[X ].
1.33 Let X denote a real-valued random variable with quantile function Q and assume that E(|X|) <
∞. Show that
1
E(X) = Q(t) dt. (1.3)
0
Let g denote a function defined on the range of X such that E[|g(X)|] < ∞. Find an expression
for E[g(X)] similar to (1.4).
1.34 Let X denote a real-valued, non-negative random variable with quantile function Q; assume
that E(X) < ∞. Fix 0 < p < 1 and let x p = Q(p) denote the pth quantile of the distribution.
Show that
1
x p ≤ E(X).
1 − p