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 ?
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See Theorem 1 given in Moschopoulos (1985) for the distribution of a sum of independent gamma variables. You can extend this result using the scaling property for linear combinations.
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1$\begingroup$ 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}} $ $\endgroup$ Commented Aug 23, 2010 at 20:53
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1$\begingroup$ where $C = \sum_{i=1}^n\left(\frac{\beta_1}{\beta_i}\right)^{\alpha_i}$ $\endgroup$ Commented Aug 23, 2010 at 20:59
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1$\begingroup$ And $\gamma_k = \sum_{i=1}^n\alpha_i\frac{(1 - \frac{\beta_1}{\beta_i})^k}{k}$ $\endgroup$ Commented Aug 23, 2010 at 21:19
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1$\begingroup$ with $\rho = \sum_{i=1}^n\alpha_i$ $\endgroup$ Commented Aug 23, 2010 at 21:19
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1$\begingroup$ @Gong-Yi: nice, thanks for updating with the answer! $\endgroup$– arsCommented Sep 14, 2010 at 1:32