[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], [math]Y[/math], and [math]Z[/math] be independent random variables

with

[[math]] f_X(x) = f_Y(x) = f_Z(x) = \left \{ \begin{array}{ll} 1, & \mbox{if $0 \lt x \lt 1,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]

Suppose that [math]W = X + Y + Z[/math]. Find [math]f_W[/math] directly, and compare your answer with that given by the formula in Example. Hint: See Example.