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Math Help - Compute E[X] and P[Y>2]

  1. #1
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    Compute E[X] and P[Y>2]

    The joint probability density function of X and Y is given by

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

    Compute E[X] and P[Y>2]
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  2. #2
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    Quote Originally Posted by Stats View Post
    The joint probability density function of X and Y is given by

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

    Compute E[X] and P[Y>2]
    As with most of the questions you have asked, apply the basic definitions:

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

    \Pr(Y > 2) = \int_{2}^{+ \infty} \int_{0}^{+ \infty} e^{-(x+y)} \, dy \, dx.

    Now integrate (it is assumed that you have the necessary calculus background).
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  3. #3
    MHF Contributor matheagle's Avatar
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    By inspection the two rvs are independent.

    X is an exponential with mean 1, hence EX=1, same for Y.

    Since \int_0^{\infty} e^{-x} \, dx=1 you only need to compute \int_2^{\infty}e^{-y} \, dy =e^{-2}.
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