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
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 $$b\cdot dx=du$$