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I'm reading about the Gamma distribution, but I'm struggling to understand what these parameters mean in a canonical sense. It was my belief that as Beta is to Binomial, Gamma is to Poisson. However, now I'm not so sure that this hits the mark.

Beta parameters can have a very tangible interpretation, a successes and b failures. However, I'm at a loss on what the scale and shape parameters of the Gamma distribution represent or do.

According to wiki, when shape, k, is a positive integer, Gamma PDF can be understood as the sum of iid exponential rvs with the same mean. And according to numpy's documentation, Gamma PDF is used frequently for modeling time to failure of electronic parts (which imo seems analogous to the geometric distribution but through the lens of time not binary events.)

So my understanding of the Gamma PDF really hasn't congealed; the parameters and PDF just seem very malleable. Any intuitions about the parameters represent as "levers" in the model?

Edit: I've seen this Q/A, however, I don't feel like it fully addresses my question and parameter interpretations. I commented on Neil G's answer, which I've copied below.

So, if an event such as time for a webpage to process a hit request was exponentially distributed, Gamma might be used to model the time it takes to process x hit requests? (Assumes only 1 hit could be processed at a time, which is an illustrative assumption but not a practical/plausible one.)

Is this example valid?

jbuddy_13
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    What the parameters "represent or do" depends on what you are using the Gamma distribution to model. – whuber Oct 08 '21 at 20:01
  • Could you point me in the direction of a semi-exhaustive list of examples? I know about as much about Gamma can model as what it can't. – jbuddy_13 Oct 08 '21 at 20:21
  • I doubt such a list exists, because Gammas show up everywhere--either as a consequence of probabilistic or physical theories, or because they might be convenient models. – whuber Oct 08 '21 at 20:59

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There is a correspondence for several pairs of distributions by means of two distributions relating to the same process.

  • one distribution describes the number of counts given some fixed time
  • the other distribution describes the (waiting)-time given some fixed number of counts occured.

The correspondence between the binomial distribution and the beta distribution is explained very well in this answer. In that answer the beta distribution is described as an order distribution, the position of the $k+1$-th event among $n+1$ events whose timing follows a uniform distribution. But you can also view this as the waiting time until the $k+1$-th event occurs.

For the Gamma distribution and the Poisson distribution you have a similar correspondence between 'waiting time' and 'number of events'. The Poisson distribution describes the number of events within some time/space, the Gamma distribution describes the time/space for a given number of events.

$$f(t) = \frac{\beta^k}{(k-1)!} t^{k-1} e^{\beta t}$$

The parameter $k$ relates to the number of events, $t$ is the time that we need to wait untill $k$ events occured, $\beta$ is a parameter relating to how fast the events occur (the rate in the Poisson process).

Sextus Empiricus
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    Another correspondence between Poisson and gamma is that the gamma distribution is the [conjugate prior](https://en.m.wikipedia.org/wiki/Conjugate_prior) for the Poisson distribution. Besides that, as Whuber commented as well, the gamma function occurs in many other places because the product $x^n e^{-x}$ can relate to many things. For instance it is als a general form of the chi-squared distribution. The explanation of parameters in this answer is solely about the relationship with the Poisson distribution. – Sextus Empiricus Oct 08 '21 at 21:37