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## Determining Limiting Distribution

1. The problem statement, all variables and given/known data

Let Xn have a gamma distribution with parameters alpha = n, and beta, where beta is not a function of n. Let Yn = Xn/n. Find the limiting distribution of Yn.

2. Relevant equations

3. The attempt at a solution

$$E(e^{tY_n}) = E(e^{t\frac{X_n}{n}})$$

I'm stuck here. I know that the MGF of a gamma distribution (in this case) is $$(1-\beta t)^{-n}$$

I'm not sure with what to do about the 1/n in the MGF. Would it be just $$\frac{(1-\beta t)^{-n}}{n}$$

Any help would be appreciated.
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