TITLE: Distribution of sum of exponential variables with different parameters
QUESTION [8 upvotes]: We have $k$ independent random variables with exponential distribution ($T_1, T_2, \ldots , T_k$), parameters of random variables are ($\lambda,\frac{\lambda}{2},\frac{\lambda}{3},\ldots,\frac{\lambda}{k}$), what is the distribution of new variable $T = T_1 + T_2 + \cdots + T_k $

REPLY [4 votes]: you can use the main result here applied to the sequence $\lambda, \lambda/2,...,\lambda/k$, which for the general case states that 
$$ h_{X_1,.,X_n}(x) = \left[ \prod_{i=1}^n \lambda_{i} \right] \sum_{j=\
1}^n \frac{e^{-\lambda_i x}}{\prod_{k \neq j}^n(\lambda_i-\lambda_j)} $$