Say I have a set of $n$ random variables, $\lbrace X_1, ..., X_n \rbrace$, each with distribution $f_{X_i}(x \mid \theta_i)$. What is $\mathbb{P}\left[\min(X_1, ..., X_n) = X_j\right]$?
I assume this will require some convolution of probabilities as this is how I would do it for $n=1, 2, 3$ but I can't see how to generalise for arbitrary $n$.