A monotonicity property of variances

Abstract

We prove that variances of non-negative random variables have the following monotonicity property: For all $0 < r < s \le 1$, and all $0 \le X \in L^2$, we have $\operatorname{Var}(X^r)^{1/r} \le \operatorname{Var}(X^s)^{1/s}$. We also discuss the real valued case.