I have Z and X which are two random variables with density:
$f_Z(z) = 3(1-z)^2\mathbb{1}_{[0,1]}(z)$
$f_X(x) = 6x(1-x)\mathbb{1}_{[0,1]}(x)$
I want to find $f_{Z \vert X = x}(z)$, but to do that I have to calculate the joint density which is unknown. (The two random variables are not independent). Is there another way? If not, how do I calculate the joint density?
EDIT: The random variable Z was defined as: $$ Z = XU $$ with $f_U(u) = \mathbb{1}_{[0,1]}(u)$