In this page, we show that the sum of two gamma-distributed random variables, Y:=X1+X2, also follows the gamma distribution: i.e. the gamma distribution has the reproductive property.
Reproductive Property of the Gamma Distribution
If the two random variables and follow the gamma distributions and :
then the sum of the two random variables follows the gamma distribution :
Preparation for the Proof
If we transform the variable in the left side of the above equation (3) into , then we obtain , , and , thus
The integration on the last part of the above equation (4) is the beta function, which is written as
The gamma distribution with the integer and the parameter implies the Erlang distribution . The proof in this page also holds in the Erlang distribution. [Reproductive Property of the Erlang Distribution (in JPN) ]