Density function of conditional random variable without knowing joint distrbution.

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 $$f_U(u) = \mathbb{1}_{[0,1]}(u)$$

• Without knowing anything about the dependence, it doesn't seem that there is enough information to solve the problem. – Greg Martin Jan 15 at 20:12
• @GregMartin I see, I will edit and add more information. – qcc101 Jan 15 at 20:16
• Again you have to tell us the relationship between $X$ and $U$, otherwise we cannot be certain. Assume you means they are independent. Then the question is easy - you do not need to do any tedious derivation - the conditional distribution of $Z$ given $X = x$ is simply $xU \sim \text{Uniform}(0, x)$. – BGM Jan 16 at 3:23
• Yes sorry about that, they indeed are independent. – qcc101 Jan 16 at 6:31