We have six exponentially distributed random variables: $X_1, X_2, X_3, Y_1, Y_2, Y_3$ with mean $\alpha_1, \alpha_2, \alpha_3, \beta_1, \beta_2, \beta_3$. We want make pairwise selection: $X_i,Y_i$, where $i = \{ 1,2,3\}$. Thus, we can either choose ($X_1$ and $Y_1$) or ($X_2$ and $Y_2$) or ($X_3$ and $Y_3$). We want to make sure that the pair selected has maximum of the minimum values.
Selection criteria: Select the maximum of the $\min(X_i, Y_i)$.
I would like to find the probability distribution $P(X_i > \tau, Y_i > \tau)$. Thank you.