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Consider two normalized gamma distribution functions $\frac{\Gamma(x,y)}{\Gamma(x)}$ and $\frac{\Gamma(nx,ny)}{\Gamma(nx)}$ where $n$ is a positive integer value. Is there any relationship between the two functions? I mean, can I write the second one as a function of the first one.

Thanks,

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  • $\begingroup$ " Is there any relationship between the two functions? I mean, can I write the second one as a function of the first one." What do you mean by relationship? Are you looking for a specific type of relationship? You can always relate any two distributions via the quantile function and the cumulative distribution function (the inverse quantile function). $\endgroup$
    – Sextus Empiricus
    Commented Oct 5, 2020 at 17:18

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No, there is no known relationship. Take a simple case of just trying to compare ${\Gamma(\alpha,x)}$ and ${\Gamma(2\alpha,x)}$.

There is no known duplication formula for the upper incomplete gamma function, so neither of these can be easily related to the other. See here for a similar question: https://math.stackexchange.com/questions/1942494/relationship-between-the-incomplete-gamma-function-of-2a-and-a?rq=1

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