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Math Help - bivariate marginal density function

  1. #1
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    bivariate marginal density function

    find the marginal density function for X and Y.

    f(x,y) = \left\{ \begin{array}{rcl}<br />
x+y & \mbox{for} & 0 \leq x \leq 1, 0 \leq y \leq 1 \\ <br />
0 & \mbox{for} & \mbox{other}<br />
\end{array}\right.

    so far I have

    f(x) = \left\{ \begin{array}{rcl}<br />
\int_{0}^{1} x+y \ dy & \mbox{for} & 0 \leq x \leq 1 \\ <br />
0 & \mbox{for} & \mbox{other}<br />
\end{array}\right.

    = f(x) = \left\{ \begin{array}{rcl}<br />
xy+\frac{y^2}{2} \ \bigg{|}^{1}_{0} & \mbox{for} & 0 \leq x \leq 1 \\ <br />
0 & \mbox{for} & \mbox{other}<br />
\end{array}\right. = f(x) = \left\{ \begin{array}{rcl}<br />
x+\frac{1}{2} & \mbox{for} & 0 \leq x \leq 1 \\ <br />
0 & \mbox{for} & \mbox{other} \end{array}\right.

    is this correct?

    And how would you find:
    P \left( X \geq \frac{1}{2} \bigg{|}Y \geq \frac{1}{2} \right) ?
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  2. #2
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    Your density for X is ok. The density for Y will be the same.

    Well, for P\left(X\geq\frac{1}{2}\bigg{|}Y\geq\frac{1}{2}\ri  ght).

    You have to apply: \frac{P\left(X\geq\frac{1}{2} , Y\geq\frac{1}{2} \right)}{P \left(Y \geq \frac{1}{2} \right)}.

    So, you have: \frac{\int_{\frac{1}{2}}^{1} \int_{\frac{1}{2}}^{1} x+y \ dy \ dx}{\int_{\frac{1}{2}}^{1} y+\frac{1}{2} \ dy}

    Hope this helps.

    Regards,

    Federico.
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