The joint probability distribution is given by,

$\displaystyle f_{(X,Y)}(x,y) = 2$ if $\displaystyle 0 \leq x \leq y \leq 1$, 0 otherwise.

(i) Determine the marginal density functions of X and Y. Check that it is indeed a pdf.

(ii) Calculate the means $\displaystyle \mathbb{E}X$ and VarX.

(iii) Find the conditional density of X given Y = y for $\displaystyle 0 \leq y \leq 1$