Help with probability please?

Question

Suppose (X,Y) has joint density
f(x,y) = 2/x * e^(-2x) for 0

Answer 1

You need the density function f_X(x) for X. You get this by integrating the joint density f(x,y) with respect to y for all y; since f(x,y) is nonzero only for 0 < y < x, this amounts to integrating 2/x * e^(-2x) with respect to y from 0 to x. Note that the condition 0 < y < x implies that x > 0. Then the conditional distribution of Y given that X = x is

f(x,y)/f_X(x)

(which of course "inherits" the restrictions on y and x coming from f(x,y) and f_X(x)).