Additivity of independent gamma distributions with a common rate
= Additivity of independent gamma distributions with a common rate
If independent random variables $X_i$ have <gamma distributions> with shapes $a_i$ and one common rate $r$, then
$$
\sum_{i=1}^nX_i\sim\operatorname{Gamma}\left(\sum_{i=1}^na_i,r\right).
$$
Their <Laplace transforms of nonnegative random variables> multiply to $(r/(r+s))^{\sum_i a_i}$, which is the transform of the displayed gamma distribution; uniqueness of the Laplace transform proves the claim.