[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 that [math]X[/math] and [math]Y[/math] are independent and [math]Z = X + Y[/math]. Find

[math]f_Z[/math] if

• [[math]] f_X(x) = \left \{ \begin{array}{ll} \lambda e^{-\lambda x}, & \mbox{if $x \gt 0,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]
  \smallskip
  [[math]] f_Y(x) = \left \{ \begin{array}{ll} \mu e^{-\mu x}, & \mbox{if $x \gt 0,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]
• [[math]] \ \ \ f_X(x) = \left \{ \begin{array}{ll} \lambda e^{-\lambda x}, & \mbox{if $x \gt 0,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]
  \smallskip
  [[math]] f_Y(x) = \left \{ \begin{array}{ll} 1, & \mbox{if $0 \lt x \lt 1,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]