What does it mean in real-world terms that the Chi-Squared distribution is a special case of the Gamma distribution?

Question

Among other applications, a gamma distribution answers the question: "If the average time (or other quantity) between events is β, what is the probability that x time will elapse before α events occur?"

The exponential distribution is a special case of the gamma distribution where α = 1. The question is easily rephrased as: "If the average time before an event occurs is β, what is the probability that x time will elapse before the event occurs?"

My textbook defines the chi-squared distribution as a special case of the gamma distribution where α = v/2 and β = 2 where v is a positive integer, but the significance of these numbers is not clear to me. Can the question in the first paragraph be rephrased to fit the chi-squared distribution in the same way I rephrased it to fit the exponential distribution?

Answer 1

The $\chi^2_n$ distribution can be seen as a $\Gamma(\frac{\nu}{2},2)$, which is called the scale parametrisation of the $\Gamma$-distribution. There is, however, also a rank interpretation. In that case the $\chi^2_n$ distribution can be seen as a $\Gamma(\frac{\nu}{2},\frac{1}{2})$, which I think lends itself more easily to a rephrasing of your question. Your question would then become:

Given $\nu$ independent processes, each with succes rate $\frac{1}{2}$, how long would it take before exactly half of these processes ($\frac{\nu}{2}$) yield a succes?