Handbook of Probability

Chapter Fourteen

Appendix B: Inequalities Involving Random Variables and Their Expectations

In this appendix we present specific properties of the expectation (additional to just the integral of measurable functions on possibly infinite measure spaces). It is to be expected that on probability spaces we may obtain more specific properties since the probability space has measure 1.

Proposition 14.1 (Markov inequality) Let Z be a r.v. and let be an increasing, positive measurable function. Then

Thus

for all g increasing functions and c > 0 .

Proof: Take λ > 0 arbitrary and define the random variable

Then clearly

and taking the expectation, we get

Example 14.1 Special Cases of the Markov Inequality

If we take g(x) = x an increasing function and X a positive random variable, then we ...