[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_1[/math], [math]X_2[/math], \dots, [math]X_n[/math] be a sequence of independent random

variables, all having a common density function [math]f_X[/math]. Let [math]A = S_n/n[/math] be their average. Find [math]f_A[/math] if

• [math]f_X(x) = (1/\sqrt{2\pi}) e^{-x^2/2}[/math] (normal density).
• [math]f_X(x) = e^{-x}[/math] (exponential density). Hint: Write [math]f_A(x)[/math] in terms of [math]f_{S_n}(x)[/math].