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 $\mathrm{Var}(X^r{)}^{1/r}\le \mathrm{Var}(X^s{)}^{1/s}$. We also discuss the real valued case.

References in corpus (3)

• Concentration of the ratio between the geometric and arithmetic means
• Selfimprovemvent of the inequality between arithmetic and geometric means
• Sharp bounds for the difference between the arithmetic and geometric means