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I need some advice to how I would begin my integration for the following problem (as in, I do not understand the region of integration): If we have a joint pdf, say $f(x,y)=4e^{-2x-2y}$ for $x,y>0$ and $A=\min(X,Y)$, how do we find $P(A>a)$ for $a>0$? My idea is to integrate $\int_w^x\int_w^y f(x,y) \, dy \, dx$. But this does not make sense because the bounds need to be restricted to a. Any suggestions?

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Hint: $P(A >a)=P((X>a)\cap (Y>a))$. Either do the double integral with $x$, $y$ each going from $a$ to $\infty$. Or else note that from the joint density function we can deduce that $X$ and $Y$ are independent, and each has exponential distribution parameter $2$. Thus $P((X>a)\cap (Y>a))=P(X>a) P(Y>a)$, and we can avoid the double integral. (Not that the double integral is difficult, since $4e^{-2x-2y}=2e^{-2x}2e^{-2y}$.)

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