In the following question a sample of random variables are given that are independent and identically distributed whereby $X_1, …, X_n \sim f_X$. It can be assumed that the random sample is from a $N(\mu,\sigma^2)$ population.
An estimator for the population mean is given as
$$\hat{\mu}_n = \frac{1}{n-1}\sum_{i=1}^n X_i$$
We are asked to use moment generating functions to define the sample distribution of $\hat{\mu}_n$
I know that the MGF for a normal distribution $N(\mu,\sigma^2)$ is
$$M(t)=\exp(\mu t+ \frac{\sigma^2 t^2}{2})$$
Is it simply a matter of taking the formula for the estimator $\hat{\mu}_n = \frac{1}{n-1}\sum_{i=1}^n X_i$ for $\mu$ and calculating the variance of the estimator and plugging both of those into the formula for $M(t)$ and simplifying?
Sorry, not sure if I'm on the right track. Any pointers much appreciated.
Best Answer
Your proposed strategy assumes $\hat{\mu}_n$ is Normal, which you might know already, but it's often proved by computing its MGF, which is the point of the problem. So a solution that requires less prior knowledge notes$$\Bbb E^{t\hat{\mu}_n}=M_{X_1}^n(t/(n-1))=\exp\left(\frac{n\mu t}{n-1}+\frac{n\sigma^2t^2}{2(n-1)^2}\right).$$