Sum of Gamma distribution with different scale.

Question

Let, $X_i$~$exp(\lambda_1)$ and $Y_i$~$exp(\lambda_2)$ iid for i = 1, 2, 3, ....
Define the r.v, $Z_1^k = \sum_{i=1}^{k}(X_i + Y_i)$ and $Z_2^k = \sum_{i=1}^{k}(X_i + Y_i) + X_{k+1}$ for k=1, 2, 3, ....
Find the pdf(or cdf) of $Z_1^k$, $Z_2^k$.

Answer 1

In general the Laplace transform of $Z=X_1+X_2+\cdots+X_n$ where the $X_i$ 's are exponential with mean $\lambda_i$ are independent and all $\lambda_i$ distinct is $$\prod_{\i=1}^n\frac{1}{1+\lambda_i s}=\sum _{\i=1}^n\frac{A_i}{1+\lambda_i s}$$ which implies that the density of $Z$ is $\sum_{i=1}^n\frac{A_i}{\lambda_i}e^{-z/\lambda_i}.$