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My book defines the gamma density as the following: $$f_X(x)=\lambda (\lambda x)^{n-1}e^{-\lambda x}/(n-1)!$$ And has the moment generating function of this density as $\frac{\lambda}{\lambda +t}$. Is this a typo, as from this solution and my own computation I think the MGF for this form of the gamma density should be $(\frac{\lambda}{\lambda - t})^n$?

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Yes, you are right.

For that special type of gamma distribution, another name for it is Erlang distribution.

It is the formula of the sum of $n$ independent exponential distributions, hence we need to raise the power.

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