Joint probability density function of two random variable X and Y

Question

I have the joint probability density function :

f(x,y)= 6(1-y) for 0< x < y <1, 0 else

I have to derive the marginal distribution of X and Y, but I don't see how to proceed.

I also have to found the conditional probability density function of X given Y=y, again I don't see how to begin the problem.

Answer 1

Use the definitions to find the marginals

$$f_X(x)=\int_{-\infty}^\infty f(x,y)dy,\qquad x\in S_1$$

and

$$f_Y(y)=\int_{-\infty}^\infty f(x,y)dx,\qquad y\in S_2$$

Just integrate them!!