Page 180 - Probability and Statistical Inference

3. Multivariate Random Variables 157

g(y) = 5/3y –8/3 I(1 < y < ∞). Now, by direct integration, one can check that
E[Y ] is finite but E[Y ] is not. Suppose that we want to get an upper bound
4/3
2
2
for |E[XY]|. The Cauchy-Schwarz inequality does not help because E[Y ] is
not finite even though E[X ] is. Let us try and apply the Hölder’s inequality
2
with r = 4 and s = 4/3 in order to obtain

Note that this upper bound is finite whatever be the nature of dependence
between these two random variables X and Y. !

3.9.6 Bonferroni Inequality
We had mentioned this inequality in the Exercise 1.3.5, part (ii), for two events
only. Here we state it more generally.
Theorem 3.9.10 (Bonferroni Inequality) Consider a sample space S
and suppose that ß is the Borel sigma-field of subsets of S. Let A , ..., A be
k
1
events, that is these belong to ß. Then,

Proof In Chapter 1, we had proved that

so that we can write

This is valid because P(A ∪ A ) ≤ 1. Similarly, we can write
1 2

So, we first verified the desired result for k = 2 and assuming this, we
then proved the same result for k = 3. The result then follows by the
mathematical induction. ¢
The Exercise 3.9.10 gives a nice application combining the Tchebysheff’s
inequality and Bonferroni inequality.