Page 198 - Probability and Statistical Inference

3. Multivariate Random Variables 175

3.9.8 First show that f(x) = x for x ∈ ℜ is a convex function. Next,
–1
+
suppose that X is a real valued random variable such that P(X > 0) = 1 and
E[X] is finite. Show that
(i) E[1/X] > 1/E[X];
(ii) Cov(X, 1/X) < 0.
3.9.9 Suppose that X and Y are two random variables with finite second
moments. Also, assume that P(X + Y = 0) < 1. Show that

{Hint: Observe that (X + Y) ≥ |X|{|X + Y|} + |Y|{|X + Y|}. Take expectations
2
throughout and then apply the Cauchy-Schwarz inequality on the rhs.}
3.9.10 Suppose that X , ..., X are arbitrary random variables, each with
n
1
zero mean and unit standard deviation. For arbitrary ε(> 1), show that

{Hint: Let A be the event that , i = 1, ..., n. By Tchebysheff’s
i
2 –1
inequality, P(A ) ≥ 1 – (nε ) , i = 1, ..., n. Then apply Bonferroni inequality,
i
–2
namely, ≥ n{1 – (nε ) } – (n–1) = 1 – ε .}
2 –1
3.9.11 Suppose that Y is a random variable for which E[Y] = 3 and E[Y ]
2
= 13. Show that P{–2 < Y < 8} > 21/25. {Hint: Check that V[Y] = 4 so that
P{–2 < Y < 8} = P{|Y – 3| < 5} > 1 – 4/25 = 21/25, by Tchebysheff’s
inequality.}