Note that I have taken a Probability Theory class (but only briefly mentioned that gamma distribution is the sum of exponential distributions) before, so somewhat rigorous explanations are fine for me. I have used generating functions before, but very limited.
Let $\forall{i},T_i$ be distributed as $exp(\lambda_i)$. Then let $T= \min\{T_1,...,T_n\}$, then $T$ has an exponential distribution with $\lambda = \sum_i^n{\lambda_i}$
Also, we know that $\sum_i^n{T_i}$ is exactly the gamma distribution with parameters $(n,\lambda)$.
My question is very specific (albeit I would be interested in a general proof too): Supposing we have $G_1,G_2$, both have gamma distributions with parameter $(2,\lambda)$, and we are interested in $G:=\min\{G_1,G_2\}$, is there an analog as to the exponential case?