Undergrad The normalizing constant in a gamma distribution

[Trollfaz]

Wikipedia and some of my simpler statistics courses present the gamma distribution with shape $\alpha$ and rate $\lambda$ as

$$P(X=x)=x^{\alpha-1}\lambda^\alpha e^{-\lambda x} \Gamma(\alpha)$$

Where $\Gamma(\alpha)$ is a value in terms of alpha to normalize the PDF from the space of x from 0 to $\infty$ (i.e make it's integral in the space 1).
I have derived that for integral values of $\alpha$,
$$\Gamma(\alpha)=\frac{1}{(\alpha-1)!}$$
But how about non integral values of alpha? How do I get the $\Gamma$ constant?
https://en.m.wikipedia.org/wiki/Gamma_distribution

 

[Trollfaz]

Someone please help with the LaTex it's not working thanks

 

[FactChecker]

I think you derived that equation for Gamma wrong. It is a well known extension of the factorial: $\Gamma(n)=(n-1)!$ for positive integers $n$, and in general, $\Gamma(z)=\int_0^\infty t^{z-1}e^{-t} dt$ for $z$ in the right half of the complex plane. (see https://en.wikipedia.org/wiki/Gamma_function )
There are tables and computer functions to get values.

 

[Trollfaz]

I think you derived that equation for Gamma wrong

Both your formula and mine results in the same final expression
$$P(X=x)=e^{-\lambda x}\lambda^n x^{n-1}/(n-1)!$$
But what if n is non integral or is it not possible for n to be non integral

 

[fresh_42]

Both your formula and mine results in the same final expression
$$P(X=x)=e^{-\lambda x}\lambda^n x^{n-1}/(n-1)!$$
But what if n is non integral or is it not possible for n to be non integral

By the integral definition @FactChecker gave you i post #3 or by the Wikipedia page about the Gamma function.

Here is a nice paper about the Gamma distribution:
https://ocw.mit.edu/courses/18-443-...924817d33e1ccb6f3a6b944c985d0cdb_lecture6.pdf
It directly starts with that definition.