I The normalizing constant in a gamma distribution

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The gamma distribution is defined with a normalizing constant, ##\Gamma(\alpha)##, ensuring the probability density function integrates to one over the interval from 0 to infinity. For integral values of ##\alpha##, it is derived that ##\Gamma(\alpha) = \frac{1}{(\alpha-1)!}##. The discussion raises questions about the gamma function's application for non-integral values of ##\alpha##, noting that it is an extension of the factorial function. The integral definition of the gamma function, ##\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt##, applies for non-integral values as well. Resources such as tables and computer functions are available to obtain gamma values for various inputs.
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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
 
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Someone please help with the LaTex it's not working thanks
 
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.
 
FactChecker said:
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
 
Trollfaz said:
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.
 
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