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[min(X,Y)\geq 2]

chillywilly12a 2021-05-28 Answered
Random variables X and Y have joint PDF
fX,Y(x,y)={12e(3x+4y), x0,y00, otherwise
Find P[min(X,Y)2]
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Expert Answer

nitruraviX
Answered 2021-05-29 Author has 101 answers

The value of P[min(X,Y)2] is obtained as given below:
P[min(X,Y)2]=P[2min(X,Y)]
=P(2X,2Y)
=P(X2,Y2)
=[1P(X<2)][1P(Y<2)]
P(X<2)=023e3xdx
=3[e3x3]02
=1e6
=0.9975
1-P(X<2)=0.0025
f(y)=xf(x,y)dx
=012e(3x+4y)dx
=12e4y[e3x3]0
f(y)=4e4y
P(Y<2)=024e4ydy
=4[e4y4]02
=1e8
=0.9997
1-P(Y<2)=0.0003
P[min(X,Y)2]=[1P(X<2)][1P(y<2)]
=0.0025×0.0003
=0.0000075

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