A monotonicity property of variances

TL;DR

This paper proves a monotonicity property of variances for non-negative random variables, showing how the variance of powers of the variable behaves as the power changes within a specific range.

Contribution

It establishes a new monotonicity property of variances for non-negative random variables, extending understanding of variance behavior under power transformations.

Findings

Variance of X^r raised to 1/r is less than or equal to that of X^s raised to 1/s for 0<r<s≤1.

The property also extends to real-valued variables.

Provides theoretical proof of the variance monotonicity property.

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.