# Math Help - Statistic Problem

1. ## Statistic Problem

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)

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

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