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I understand that a chi-squared distribution is a special case of the gamma distribution. However, I find claims of "the math just works out" to be an unhelpful in remembering or understanding the relationship.

Is there a satisfying way to relate these two distributions? For example, is there a way to connect the Poisson process interpretation of the gamma to the "sum of squares of iid normals" interpretation of the chi-squared? Why does the gamma have anything to do with Gaussian r.v.s that has been squared?

For possible reference, some intuition I currently have:

  • I intuit the gamma distribution as the waiting time for the $k$-th arrival in a Poisson process. This supports the idea that independent gamma r.v.s (with same rate parameter) can be summed to another gamma distribution.
  • A gamma distribution with a large shape parameter can be thought of as the sum of many independent gamma r.v.s with smaller shape parameters. By CLT, the gamma converges to a normal distribution as the shape parameter grows. (Same deal with the chi-squared distribution.)
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    $\begingroup$ I don't know of (and couldn't find) any relationship between the limit process that produces Gaussians (the CLT) and that produces a Poisson process (superpositiong and thinning). $\endgroup$ Oct 3 '20 at 3:55
  • $\begingroup$ @ThomasLumley Perhaps this is of interest to you as an answer to this supposed mystery. $\endgroup$
    – Carl
    Oct 17 at 7:48
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Intuition has to be trained through arduous application to become other than misleading. There are too many implied questions here for a single post. However, addressing those here does provide a series of links summarizing some of the properties of the gamma distribution, so the implied questions posited may have some value to the potential reader.

Q1 Chi-squared and gamma distribution

A1 See end of answer here, and even more explicitly here. In particular in that latter answer, note the following statement: "Unique properties are scattered around all over Mathematics, and most of the time, they don't reflect some "deeper intuition" or "structure" - they just exist (thankfully)."

Q2 "is there a way to connect the Poisson process interpretation of the gamma"

A2 See answer No, there is no Poisson process interpretation. Poisson is a subset of gamma for positive integers only. Thus, there is a gamma simplification that leads to Poisson not the obverse.

Q3 An implied question: "I intuit the gamma distribution as the waiting time for the 𝑘-th arrival in a Poisson process. This supports the idea that independent gamma r.v.s (with same rate parameter) can be summed to another gamma distribution."

A3 Backwards again. Although the special case of same rate parameter has closure under convolution (sum of rv's), this is not the case without having the same rate parameter, where the sum (convolution) of two gamma distributions is not closed by a gamma distribution. For what it actually is see this link for the sum of two gammas, and for more than two see this other link. Second, as a Poisson distribution is a gamma distribution subset one can at most "suspect" that a gamma distribution might have something to do with wait times. What is lacking is any such determination for anything beyond that special case simplification.

Statement "A gamma distribution with a large shape parameter can be thought of as the sum of many independent gamma r.v.s with smaller (Sic, iff identical) shape parameters." It can, but to what end?

Q4 "By CLT, the gamma converges to a normal distribution as the shape parameter grows."

A4 The shape does but the mean would grow without bound to do so. To maintain stationarity, a lot more has to be done. See this answer.

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