The joint density of the random variables X and Y is given by f(x,y)={8xy0≤x≤1,0≤y≤x0otherwise Find...

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