# How to scale gamma distribution to longer interval?

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:

1. Set the second parameter of the Gamma distribution to be "10 years", i.e. Gamma(2400, 10)
2. 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?

• Note that "number of hundred degree days in a year" is a count (i.e. necessarily discrete), and one that has a strict upper bound (you can't have more hundred degree days than days in a year), so it cannot possibly have any Gamma distribution, despite the statement otherwise there. There has to be truncation and rounding (or similarly, some other way of dealing with the distinction between a continuous variable with semi-infinite support and a discrete variable with bounded support), leaving you with not-actually-a-gamma. – Glen_b Jul 20 '20 at 5:11

Assuming the years are independent, the number of 100-degree days in Austin over the next 10 years will simply be $$10\lambda$$. The number of hot days sum up for each year over the 10 years.
Now if $$\lambda\sim\mathrm{Gamma}(18,0.75)$$, then $$10\lambda\sim\mathrm{Gamma}(180, 0.75)$$ (see here for a reference). In general, if $$X_i\sim\mathrm{Gamma}(k, \theta)$$, then $$\sum_{i=1}^{N}{X_{i}}\sim\mathrm{Gamma}\left(Nk, \theta\right)$$.