Statistic Problem

zorro · #1 $Old$ January 7th, 2009, 05:13 AM

Suppose that the joint density of X and Y is given by

$f(x,y) = \begin{cases} exp[-(x+y)], & 0 \leqslant x,y< \infty \\ 0, & otherwise \end{cases}$

Find E[X] and P(Y>1)

mr fantastic · #2 $Old$ January 7th, 2009, 05:45 AM

Quote:

Originally Posted by zorro $View Post$

Suppose that the joint density of X and Y is given by

$f(x,y) = \begin{cases} exp[-(x+y)], & 0 \leqslant x,y< \infty \\ 0, & otherwise \end{cases}$

Find E[X] and P(Y>1)

$E(X) = \int_{x=0}^{+\infty} x \, \left( \int_{0}^{+\infty} e^{-(x+y)} \, dy\right) \, dx$

$= \int_{x=0}^{+\infty} x e^{-x} \, \left( \int_{0}^{+\infty} e^{-y} \, dy\right) \, dx$ .

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$\Pr(Y > 1) = \int_{y=1}^{+\infty} \left( \int_{0}^{+\infty} e^{-(x+y)} \, dx\right) \, dy$

$= \int_{y=1}^{+\infty} e^{-y} \, \left( \int_{0}^{+\infty} e^{-x} \, dx\right) \, dy$ .