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Thread: bivariate marginal density function

  1. #1
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    bivariate marginal density function

    find the marginal density function for $\displaystyle X$ and $\displaystyle Y$.

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

    so far I have

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

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

    is this correct?

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

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

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

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

    Hope this helps.

    Regards,

    Federico.
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