I have this problem that has caught me off guard:

Let X be a continous random variable with PDF:

\(\displaystyle f_(xy) (x,y)= xc^2y ; 0 \leq y \leq x \leq 1 ; 0 else where\)

I computed this to be the area bounded by the line y=x, x-axis and x=1.a) Find the range

From the fact that the total area under the range must be 1 , I get c=10b) Find the constant c

I get:c) Find the marginal PDFs, fx (x) and fy(y)

\(\displaystyle f_x(x) = \int_0^x 10x^2y dy = 5x^4 ; 0 \leq x \leq 1\)

\(\displaystyle f_y(y) = \int_0^y 10x^2y dx = 10/3 y^4 ; 0 \leq x \leq 1\)

I am having trouble here, I'm not sure I understand what the question is asking. What does 'Y' represent? Am I going to be working with the marginal PDF with respect to Y, or am I working with the multi variable original function?d) Find \(\displaystyle P(Y\leq X/2)\)

Can I make the substitution y=x and solve using the marginal PDF for the variable y?

I am completely lost here. I would appreciate an explanation of what is going on.e) Find \(\displaystyle P (Y \leq X/4 | Y \leq X/2) \)