 # Random variables X and Y have joint PDF f_{X,Y}(x,y)=\begin{cases}12e^{-(3x+4y)},\ x \geq 0, y \geq 0\\0,\ otherwise\end{cases} Find P[max(X,Y)\leq 0.5] OlmekinjP 2021-05-30 Answered
Random variables X and Y have joint PDF

Find $P\left[max\left(X,Y\right)\le 0.5\right]$
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The value of $P\left[max\left(X,Y\right)\le 0.5\right]$ is obtained as given below:
$P\left[max\left(X,Y\right)\le 0.5\right]=P\left(X\le 0.5,Y\le 0.5\right)$
$=P\left(X\le 0.5\right)×P\left(Y\le 0.5\right)$
$P\left(X<0.5\right)={\int }_{0}^{0.5}3{e}^{-3x}dx$
$=3\left[-\frac{{e}^{-3x}}{3}{\right]}_{0}^{0.5}$
$=1-{e}^{-1.5}$
=0.7769
$P\left(Y<0.5\right)={\int }_{0}^{0.5}4{e}^{-4y}dy$
$=4\left[-\frac{{e}^{-4y}}{4}{\right]}_{0}^{0.5}$
$=1-{e}^{-2}$
=0.8647
$P\left[max\left(X,Y\right)\le 0.5\right]=P\left(X\le 0.5\right)×P\left(Y\le 0.5\right)$
$=07769×0.8647$
=0.6718