[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] and [math]Y[/math] be independent random variables defined

on the space [math]\Omega[/math], with density functions [math]f_X[/math] and [math]f_Y[/math], respectively. Suppose that [math]Z = X + Y[/math]. Find the density [math]f_Z[/math] of [math]Z[/math] if

• [[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} 1/2, & \;\mbox{if $-1 \leq x \leq +1,$} \\ 0, & \;\mbox{otherwise.} \end{array} \right. [[/math]]
• [[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} 1/2, & \;\mbox{if $3 \leq x \leq 5,$} \\ 0, & \;\mbox{otherwise.} \end{array} \right. [[/math]]
• [[math]] f_X(x) = \left \{ \begin{array}{ll} 1/2, & \;\mbox{if $-1 \leq x \leq 1,$} \\ 0, & \;\mbox{otherwise.} \end{array} \right. [[/math]]
  \smallskip
  [[math]] f_Y(x) = \left \{ \begin{array}{ll} 1/2, & \;\mbox{if $3 \leq x \leq 5,$} \\ 0, & \;\mbox{otherwise.} \end{array} \right. [[/math]]
• What can you say about the set [math]E = \{\,z : f_Z(z) \gt 0\,\}[/math] in each case?