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On this wikipedia page negative binomial distribution, Negative binomial was derived as integrating out the lambda from Gamma-Poisson mixture.

I tried to follow the proof step by step, but I am stuck in last step where the author used what I think is the following in the proof:

$$\frac{\Gamma(t)}{b^t}=\int_0^\infty x^{t-1}e^{-xb}dx$$

I understand Gamma function is defined as this: $$\Gamma(t)=\int_0^\infty x^{t-1}e^{-x}dx$$

But I couldn't get my head around how the former was derived and wolfram alpha is no help. My question is 1) Does this identity hold? and 2) if so why

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Yes, this identity holds. Observe that $$\frac{\Gamma(t)}{b^t}=\int_0^\infty x^{t-1}e^{-xb}dx$$ can be written (by multiplying both sides with $b^t$) equivalently as $$\Gamma(t)=b^t\int_0^\infty x^{t-1}e^{-xb}dx=b\int_0^\infty (xb)^{t-1}e^{-xb}dx$$ (integration is with respect to $x$ not $t$, so you can do it!). Now subsitute $u:=xb$ to obtain the equality, since this transformation does not change the integration limits and $$b\cdot dx=du$$

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