I'm trying to find a closed form for the hypoexponential distribution (the sum of exponentially distributed RVs with different rates). Does anybody have an idea about an approximation for this distribution? mainly if I can get rid of the product term in the expression, even by approximation.
The formula is as follows:
$f_x(x) = \sum\limits_{i=1}^{k} (\prod\limits_{j\neq i}\frac{\lambda_j}{\lambda_j-\lambda_i}) \lambda_i exp(-\lambda_i x)$