Complete Derivation of the Gamma Distribution: PDF, MGF, Mean, Variance 

In this video I discuss the gamma function and its connection to the gamma distribution. From there I derive the PDF of the gamma distribution, ending with a derivation for the MGF, Mean, and Variance.
00:49 Introduction to the gamma distribution.
04:10 Gamma Function introduction and "derivation".
15:08 Showing that the integral representation satisfies fundamental property of the factorial function.
16:37 Fractional factorials; result of analytic continuation.
19:10 Gamma Function as a shifted analytically continued factorial function.
22:28 Derivation of the PDF for the gamma distribution.
36:51 Alternate representations of the gamma distribution's PDF.
40:05 Derivation of the MGF for the gamma distribution.
48:28 Derivation of the mean and variance.
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