Can you sum a Gamma distribution (n,1/lambda) + Exponential (1/lambda*)?

Question

I can proof that the exponential distribution can be written as a Gamma (1,lambda*). And, it is also known that the sum of two Gamma is also a Gamma. But, in this case the distributions do not have the same scale parameter. How does the pdf of W=Gamma(n,1/lambda)+Gamma(1,lambda*).

Answer 1

Let $X\sim\Gamma(\alpha_{X},\lambda_{X})$ and $Y\sim\Gamma(\alpha_{Y},\lambda_{Y})$ be independent random variables. Denote by $\phi_{A}$ the characteristic function of a random variable $A$. Then, $$ \phi_{X+Y}(t)=\phi_{X}(t)\phi_{Y}(t)=\left(1-\frac{it}{\lambda_{X}}\right)^{-\alpha_{X}}\left(1-\frac{it}{\lambda_{Y}}\right)^{-\alpha_{Y}}. $$ Therefore, if $\lambda_{X}=\lambda_{Y}\equiv\lambda$, $$ \phi_{X+Y}(t)=\left(1-\frac{it}{\lambda}\right)^{-\alpha_{X}-\alpha_{Y}}. $$ That is, $X+Y\sim\Gamma(\alpha_{X}+\alpha_{Y},\lambda)$ and the PDF of $X+Y$ is $$ \frac{\lambda^{\alpha_X+\alpha_Y} x^{\alpha_X+\alpha_Y-1} e^{-\lambda x}}{\Gamma(\alpha_X+\alpha_Y)}. $$ Note that if $\lambda_X \neq \lambda_Y$, the sum $X+Y$ is not Gamma distributed.