I'm trying to gain intuition for how to scale the parameters of the Gamma distribution when the desired variance is not known. I'll make up an example:
Let $\lambda$ be the number of 100-degree days in Austin, Texas per year. From historical observation, $\lambda \sim$ Gamma(18, 0.75).
If I want to estimate the probability that there will be 250 100-degree days in Austin over the next 10 years, how do I know what Gamma distribution to use? Intuitively, I have at least two reasonable choices:
- Set the second parameter of the Gamma distribution to be "10 years", i.e. Gamma(2400, 10)
- Linearly scale the first parameter to be 10 * the number of events, i.e. Gamma(180, 0.75)
If I knew what the variance of the distribution was supposed to be, then this question would be trivial. If there is no predefined variance, is there a way of identifying which of these two distributions is more reasonable?