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

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    $\begingroup$ Without knowing anything about the dependence, it doesn't seem that there is enough information to solve the problem. $\endgroup$ – Greg Martin Jan 15 at 20:12
  • $\begingroup$ @GregMartin I see, I will edit and add more information. $\endgroup$ – qcc101 Jan 15 at 20:16
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    $\begingroup$ 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 $xU \sim \text{Uniform}(0, x)$. $\endgroup$ – BGM Jan 16 at 3:23
  • $\begingroup$ Yes sorry about that, they indeed are independent. $\endgroup$ – qcc101 Jan 16 at 6:31

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