[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]

Suppose again that [math]Z = X + Y[/math]. Find [math]f_Z[/math] if

• [[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} x/2, & \mbox{if $0 \lt x \lt 2,$} \\ 0, & \mbox{otherwise}. \end{array} \right. [[/math]]
• [[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} (1/2)(x - 3), & \mbox{if $3 \lt x \lt 5,$} \\ 0, & \mbox{otherwise}. \end{array} \right. [[/math]]
• [[math]] f_X(x) = \left \{ \begin{array}{ll} 1/2, & \mbox{if $0 \lt x \lt 2,$} \\ 0, & \mbox{otherwise}, \end{array} \right. [[/math]]
  \smallskip
  [[math]] f_Y(x) = \left \{ \begin{array}{ll} x/2, & \mbox{if $0 \lt x \lt 2,$} \\ 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?