I'm trying to find the probability distribution and expected value of the minimum of maximums of a combination of random variables. For example, say
$$X_1 \sim \mathrm{Exp}(\text{rate}=\lambda_1), X_2 \sim \mathrm{Exp}(\text{rate} = \lambda_2), X_3 \sim \mathrm{Exp}(\text{rate} = \lambda_3) $$
I can then define $W_{12} = \max(X_1,X_2)$ and $W_{13} = \max(X_1,X_3)$ and what to know the distribution and expectation of $Z = \min(W_{12},W_{13})$
Here's what I have so far:
\begin{align} F_Z(t) &= P(Z\leq t) \\ &= P(\min(W_{12},W_{13})\leq t) \\ &= P(\{w_{12} \leq t\} \cup \{w_{13} \leq t\}) \\ &= P(W_{12}\leq t) + P(W_{13}\leq t) - P(\{w_{12} \leq t\} \cap \{w_{13}\leq t\})\end{align}
Im having issues calculating $P(\{w_{12} \leq t\} \cap \{w_{13}\leq t\})$ because I know $W_{12}$ and $W_{13}$ are not independent.
How do I proceed?