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If we have a Gamma distribution and the density function is $$f_r(x)=\frac{\lambda^rx^{r-1}e^{-\lambda x}}{(r-1)!}$$ How come we write that the density function is: $$f_r(x)=\frac{\lambda^rx^{r-1}e^{-\lambda x}}{\Gamma(r)}$$ I don't understand the reasoning behind using $\Gamma (r)$ versus $(r-1)!$. Also if we are to integrate the Gamma Distribution to find a probability then is are we integrating with respect to $r$ or $x$?

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$r$ is a parameter, $x$ is the variable, so you integrate with respect to $x$. The main reason for using $\Gamma(r)$ rather than $(r-1)!$ is so that $r$ is not required to be an integer.