Let $X$ and $Y$ be two random variables. Their joint PDF is uniform in the region $0$ to $1$ (inclusive). Let $Z$ be a random variable defined as $Z = \min\{X,Y\}$. Determine $f_Z (z)$, $f_{Z\mid X}(z\mid x)$, $E[Z],$ and $E[X\mid Z=z] $
I'm currently working on $f_Z(z)$. I have $P(Z \leq z) = P(X \leq z)P(y \leq\ z)$
First question, is this even correct? I've been trying to figure out how to define the $\min(X,Y)$ requirement, and this is what I have seen repeated a few times. And if it is correct... how do I evaluate it? I know it's an integral, but what am I integrating from? I could use some conceptual help on understanding what is being asked of me.