Rearrangements of distributions on integers that minimize variance

Abstract

Which permutations of a probability distribution on integers minimize variance? Let $X$ be a random variable on a set of integers $\{x_1,\dots ,x_N\}$ such that $\mathbb{P}(X_i=x_i)=p_i$, $i\in \{1,\dots ,N\}$. Let $(p^{(1)},\dots ,p^{(N)})$ be the sequence $(p_1,\dots ,p_N)$ ordered non-increasingly. Let $X^{+}$ be the random variable defined by $\mathbb{P}(X^{+}=0)=p^{(1)}$, $\mathbb{P}(X^{+}=1)=p^{(2)}$, $\mathbb{P}(X^{+}=-1)=p^{(3)},\dots ,\mathbb{P}(X^{+}=(-1{)}^N\lfloor \frac{N}{2}\rfloor )=p^{(N)}$. In this short note we generalize and prove the inequality $\mathrm{Var}{X}^{+}\le \mathrm{Var}X$.