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?