Let (X_{1}, ldots, X_{n}) be a sequence of independent random variables where (X_{i}) has a (operatorname{Gamma}left(alpha_{i}, betaight))
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Let \(X_{1}, \ldots, X_{n}\) be a sequence of independent random variables where \(X_{i}\) has a \(\operatorname{Gamma}\left(\alpha_{i}, \betaight)\) distribution for \(i=1, \ldots, n\). Let
\[S_{n}=\sum_{i=1}^{n} X_{i}\]
Find the moment generating function of \(S_{n}\), and identify the corresponding distribution of the random variable.
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