# Thread: bivariate marginal density function

1. ## 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:
$\displaystyle P \left( X \geq \frac{1}{2} \bigg{|}Y \geq \frac{1}{2} \right)$ ?

2. 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.