Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1.
I would like to find E($\bar{Y^4})$ by using moment generating function.
The setup I have right now is the following:
E($\bar{Y^4}) = \int_{-\infty}^\infty \frac{\bar{Y}^4}{2\pi^{n/2}} e^{-\sum_{i=1}^n y_i^2 - 2\mu n\bar{Y} + n\mu^2}$
First of all, is the setup correct? If so, I am wondering how to get rid of $y_i$?