Let X denote a random variable distributed as noncentral 2 with f degrees of freedom and noncentrality parameter 2. Then X is stochastically larger than

Let Xλ denote a random variable distributed as noncentral χ2 with f degrees of freedom and noncentrality parameter λ2. Then Xλ is stochastically larger than Xλ if λ<λ

.

[It is enough to show that if Y is distributed as N(0, 1), then (Y + λ

)2 is stochastically larger than (Y + λ)2. The equivalent fact that for any z > 0, P{|Y + λ

| ≤ z} ≤ P{|Y + λ| ≤ z}, is an immediate consequence of the shape of the normal density function. An alternative proof is obtained by combining