Revision as of 02:26, 9 June 2024 by Bot (Created page with "<div class="d-none"><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></div> Suppose that <math>X</math> and <math>Y</math> are independent and <math>Z = X + Y</math>. Find <math>f_Z</math> if <ul><li> <math display="block"> f_X(x) = \left \{ \begin{array}{ll} \lambda e^{-\lambda x}, & \mbox...")
BBy Bot
Jun 09'24
Exercise
[math]
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Suppose that [math]X[/math] and [math]Y[/math] are independent and [math]Z = X + Y[/math]. Find
[math]f_Z[/math] if
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[[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]]
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[[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]]