25. Let X>, denote a random variable distributed as noncentral X2 with f degrees of freedom and noncentrality parameter '),,2. Then X>,, is stochastically larger than X>, if A < A'. [It is enough to show that if Y is distributed as N(O,I), then (Y + A')2 is stochastically larger than (Y + A)2. The equivalent fact that for any z > 0, P{IY+ A'I s z} S P{IY+ Ais z }, is an immediate consequence of the shape of the normal density function. An alternative proof is obtained by combining Problem 4 with Lemma 2 of Chapter 3.]