I have this homework question where there are 3 random variables $(U_0,U_1,U_2)$ which are independent and uniform in the interval $[-1,1]$.
I have two other random variables $(X,Y)$ defined as follows: $$X=\min(U_0,U_1)\quad,\quad Y=\min(U_1,U_2)$$
I am asked to find joint pdf of $X$ and $Y$.
I have some rough ideas but any hint to suggest how to solve this is very appreciated. Thanks a lot in advance.
For $X$, I calculated $$P(X=x)$$ as follows: $$P(X=x)=P(U_0>x|U_1=x) \,\,\, \cup \,\,\, P(U_1>x|U_0=x)$$ and since these belong to different probability spaces and $U_0$ and $U_1$ are independent: $$P(X=x)=P(U_0>x)+P(U_1>x)$$
Is it going ok so far?