Determining Limiting Distribution

[cse63146]

Homework Statement

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.

Homework Equations

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.

 

[mighty2000]

Thank you for your question. In order to find the limiting distribution of Yn, we can use the concept of the Central Limit Theorem (CLT). The CLT states that as the sample size increases, the distribution of the sample mean approaches a normal distribution with mean equal to the population mean and variance equal to the population variance divided by the sample size. In our case, the sample mean is Yn = Xn/n.

We know that the gamma distribution has a mean of alpha/beta and a variance of alpha/beta^2. In this case, as n increases, the mean of Yn approaches 1/beta and the variance approaches 1/(beta^2n). Therefore, the limiting distribution of Yn is a normal distribution with mean 1/beta and variance 1/(beta^2n).

I hope this helps. Let me know if you have any further questions.