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variance inequality

Source: VJIMC 2022 2.3

April 11, 2022
probability theoryinequalities

Problem Statement

Let x1,…,xnx_1,\ldots,x_n be given real numbers with 0<m≤xi≤M0<m\le x_i\le M for each i∈{1,…,n}i\in\{1,\ldots,n\}. Let XX be the discrete random variable uniformly distributed on {x1,…,xn}\{x_1,\ldots,x_n\}. The mean μ\mu and the variance σ2\sigma^2 of XX are defined as μ(X)=x1+…+xnn and σ2(X)=(x1−μ(X))2+…+(xn−μ(X))2n.\mu(X)=\frac{x_1+\ldots+x_n}n\text{ and }\sigma^2(X)=\frac{(x_1-\mu(X))^2+\ldots+(x_n-\mu(X))^2}n. By X2X^2 denote the discrete random variable uniformly distributed on {x12,…,xn2}\{x_1^2,\ldots,x_n^2\}. Prove that σ2(X)≥(m2M2)2σ2(X2).\sigma^2(X)\ge\left(\frac m{2M^2}\right)^2\sigma^2(X^2).
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