# What are we modelling when a gamma distribution has non-integer shape parameter

I wish to receive a clear and concise answer as to what is being modeled for a gamma distribution with non-integer shape parameter, and a more detailed derivation of its distribution function for all positive numbers (rather than simply integers).

I have derived the gamma distribution as the sum of $$n \in \{0, 1, 2, ...\}$$ exponentially distributed random variables. I also have shown via contradiction (or rather, my lecturer has), that a situation in which $$k-1$$ events occur at some given rate $$\lambda_1$$, and one event occurs at a rate $$\lambda_2 = \frac{\lambda_1}{2}$$, cannot be modelled by a gamma distribution (no matter the rate parameter, as a compound Poisson process does not follow a gamma distribution, although the two distributions are very close).

Thus, if this the two distributions are not the same, how would I derive the gamma pdf for a non-integer shape parameter?

I have read the answer here: Is there another interpretation for a Gamma distribution with non-integer shape parameter?

However this is with a rate parameter of one, and thus not general enough. All the textbooks I have merely introduce or define the distribution without any explanation or derivation, which I find ridiculous. We may be able to use this definition to prove that it is a valid pmf, however it does not prove that it has the property of modelling the things we wish to model. This is why I am searching for a derivation based on a non-integer shape parameter.

• Fact is the gamma distribution is a fairly general shape and its interpretations as a sum of exponentials, or as the composite result of events in time, only cover special cases. It's up to you whether you find that ridiculous. Nov 28, 2022 at 16:44
• You also may want to see the second answer here: mathoverflow.net/q/38821/39144 Nov 28, 2022 at 17:07
• Chi-squared variables with odd d.f. have Gamma distributions with non-integral shape parameters. If you want a mathematical derivation, look into Jacobi sums on arbitrary fields or contemplate the classical relationship between Gamma and the cosecant function.
– whuber
Nov 28, 2022 at 17:09
• @NickCox - The fact that it has a general shape is fine, however I am talking specifically about deriving the distribution based on a specific situation. Sure, it may have a general shape, but that property is not an explanation as to where it came from in the first place, nor is it an explanation as to what specifically is being modelled theoretically when we have an non-integer (+ve) shape parameter. This is the part I find very frustrating, is that I can only find derivations for an integer shape parameter, which leaves an infinite number of cases unexplained. Hence my question.
– Cai
Nov 29, 2022 at 12:43
• I like it when distributions arise from plausible models, but many have no clear motivation and some have so many that you are hard pushed to know which to believe. There are all sorts of ways, for example, to get negative binomials, or the same rose under different names. Nov 29, 2022 at 12:53

Mathematically Gamma distribution may also arise as a stationary solution of Fokker-Planck equation: $$\partial_t P(x,t)=-\partial_xJ(x,t),\\J(x,t)=\mu(x)P(x,t)- \partial_x [D(x)P(x)]$$ with some choices of parameters, like $$D(x)=D_0x$$ and $$\mu(x)=-\mu_1x+\mu_0$$, which gives $$J(x,t)=0\Rightarrow P_0(x)\propto x^{\frac{\mu_0}{D_0}-1}e^{-\frac{\mu_1}{D_0}x}$$ Such equations arise, e.g., in mathematical population genetics.