22. Suppose that X is the lifetime of a randomly selected fan used in certain types of diesel engines. Let Y be a randomly selected competing fan for the same type of diesel engines manufactured by another company. To compare the lifetimes X and Y, it is not sufficient to compare E(X) and E(Y ). For example, E(X) > E(Y ) does not necessarily imply that the first manufacturer’s fan outlives the second manufacturer’s fan.

Knowing Var(X) and Var(Y ) will help, but variance is also a crude measure. One of the best tools for comparing random variables in such situations is stochastic comparison.

Let X and Y be two random variables. We say that X is stochastically larger than Y, denoted by X ≥st Y, if for all t,

P(X > t) ≥ P(Y > t).
Show that if X ≥st Y, then E(X) ≥ E(Y ), but not conversely.
Hint: Use Theorem 6.2.