Sum of 3 independent gamma distributed random variable

Question

If $X\sim \Gamma(a_1,b)$, $Y\sim\Gamma(a_2,b)$ and $Z\sim\Gamma(a_3,b)$, how to prove that $X+Y+Z \sim \Gamma(a_1+a_2+a_3,b)$ if $X$, $Y$ and $Z$ are independent.

Answer 1

If $X\sim\Gamma(a_1,b)$, then its characteristic function is given by $\varphi_X(t)=(1-\frac{it}{b})^{-a_1}$.

By independence of $X$, $Y$ and $Z$, the characteristic function of $X+Y+Z$ is the product of the characteristic functions, i.e. for any $t\in\mathbb{R}$,

$\varphi_{X+Y+Z}(t)=(1-\frac{it}{b})^{-(a_1+a_2+a_3)}$,

and since the characteristic function characterizes the distribution, $X+Y+Z\sim\Gamma(a_1+a_2+a_3,b)$.