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In my statistics class, it was proven that the sum of independent and identically distributed random variables distributed according to an exponential distribution follows a gamma distribution using moment-generating functions. However, one thing that confuses me is how the formula for the Gamma distribution seems to come out of nowhere; i.e. the proof confirms that it is correct, but it does not seem to derive it from the exponential distribution definition.

Anon
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    The proof derives the moment generating function of a sum of i.i.d. exponentials and shows that it is the same as the m.g.f. of the gamma distribution. Why does the Gamma distribution need to be derived from anything? It's just a formula with a name attached to it; the incomplete gamma function normalized so that $\lim_{t \rightarrow \infty}F(t) = 1$. – jbowman Sep 11 '18 at 03:42
  • If we define the Gamma distribution to be the distribution of the sum of i.i.d. exponentials, then the formula $x^{k-1}e^{-x}$ seems a bit unmotivated. Sure, we see that when we calculate the integral to get the formula for the MGF of a gamma distributed random variable that it matches the formula for the sum of i.i.d. exponentials, but my question is how does one even discover the formula $x^{k-1}e^{-x}$ in the first place? – Anon Sep 11 '18 at 03:49
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    That's not how the Gamma distribution is defined; $k$ doesn't have to be an integer, for one thing. The Gamma distribution is just the incomplete gamma function (https://en.wikipedia.org/wiki/Incomplete_gamma_function) divided by the gamma function itself, with something extra to allow the random variable to be rescaled. Look down the Wikipedia page to "Regularized Gamma functions..." and the relationship is spelled out. – jbowman Sep 11 '18 at 04:00

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