[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] be a random variable with density function [math]f_X[/math].

Show, using elementary calculus, that the function

[[math]] \phi(a) = E((X - a)^2) [[/math]]

takes its minimum value when [math]a = \mu(X)[/math], and in that case [math]\phi(a) = \sigma^2(X)[/math].