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Math Help - Statistic Problem

  1. #1
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    Statistic Problem

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

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

    Find E[X] and P(Y>1)
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  2. #2
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    Quote Originally Posted by zorro View Post
    Suppose that the joint density of X and Y is given by

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

    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.

    -------------------------------------------------------------------------------------

    \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.
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