# The distribution of the linear combination of Gamma random variables [duplicate]

If $X_i\sim\Gamma(\alpha_i,\beta_i)$ for $1\leq i\leq n$, let $Y = \sum_{i=1}^n c_iX_i$ where $c_i$ are positive real numbers. Assume all the parameters $\alpha_i$'s and $\beta_i$'s are all known, what is $Y$'s distribution ?

• According to reference, the distribution function is: $g(y) = C\sum_{k=1}^\infty\frac{\delta_k y^{\rho+k-1}e^{-\frac{y}{\beta_1}}}{\Gamma(\rho+k)\beta_i^{\rho+k}}$ Aug 23, 2010 at 20:53
• where $C = \sum_{i=1}^n\left(\frac{\beta_1}{\beta_i}\right)^{\alpha_i}$ Aug 23, 2010 at 20:59
• And $\gamma_k = \sum_{i=1}^n\alpha_i\frac{(1 - \frac{\beta_1}{\beta_i})^k}{k}$ Aug 23, 2010 at 21:19
• with $\rho = \sum_{i=1}^n\alpha_i$ Aug 23, 2010 at 21:19