Trent Carpenter

2021-11-03

The joint density of the random variables X and Y is given by
$f\left(x,y\right)=\left\{\begin{array}{ll}8xy& 0\le x\le 1,0\le y\le x\\ 0& otherwise\end{array}$
Find the conditional density of X
Find the conditional density of Y

smallq9

Step 1
We have been given the joint density function of random variables X and Y as,
$f\left(x,y\right)=\left\{\begin{array}{ll}8xy& 0\le x\le 1,0\le y\le x\\ 0& otherwise\end{array}$
$=\left\{\begin{array}{ll}8xy& 0\le x\le 1,0\le y\le x\\ 0& otherwise\end{array}$
The conditional density function of X is computed as,
$P\left(X=x\mid Y=y\right)=\frac{f\left(x,y\right)}{f\left(y\right)}=\frac{8xy}{4y}=2x$
The conditional density function of Y is computed as,
$P\left(Y=y\mid X=x\right)=\frac{f\left(x,y\right)}{f\left(x\right)}=\frac{8xy}{4x}=2y$

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