[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] with support [math][a,b][/math] (see Exercise Exercise). Let [math]S_n = X_1 + X_2 +\cdots+ X_n[/math], with density function [math]f_{S_n}[/math]. Show that the support of [math]f_{S_n}[/math] is the interval [math][na,nb][/math]. Hint: Write [math]f_{S_n} = f_{S_{n - 1}} * f_X[/math]. Now use Exercise Exercise to establish the desired result by induction.