Mathematische Statistik - Übung Ergänzungsaufgabe 2 Beispiel 5

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Prove that the class fo  \Gamma \, distributions is closed under multiplication.

 f(x) = x^{k-1} \frac{e^{-\frac x \theta}}{\theta^k \Gamma(k)} \propto x^{k-1} e^{- \frac x \theta} \,


\begin{align}
f_{k, \theta}(x) g_{i, \gamma}(x)
& \propto x^{k-1} e^{- \frac{x}{\theta}} x^{i-1} e^{- \frac{x}{\gamma}} \\
& \propto x^{(k+i-1) - 1} e^{- \frac{\gamma x  - \theta x}{\gamma \theta}} \\
& \propto f_{k_i-1, \frac{\theta \gamma}{\theta - \gamma}}(x)
\end{align}
 \,