Let $Z_1, Z_2, Z_3$ independent RVs with exponential distribution$(\lambda = 1)$.
Let $X = \min \{Z_1,Z_3\}$ and $Y =\min \{Z_2,Z_3\}$.
I need to find $P(X>x,Y>y)$, $P(X=Y)$ and $E[X|Y=2]$.
I know that $P(X>x) = P(Z_1>x)P(Z_3>x)$ and $P(Y>y) = P(Z_2>y)P(Z_3>y)$. However I get stuck here because it seems X are Y are not independent.
How should I approach this? Should I be conditioning for all the possible orders of $Z_1, Z_2, Z_3$ (ex: $Z_1>Z_2>Z_3, Z_1>Z_3>Z_2$, etc)?