Suppose that $\displaystyle Y_1,Y_2,...,Y_n$ are a random sample from a normal distribution with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$. Use MGFs to deduce the PDF of $\displaystyle \bar{Y}$.
Are $\displaystyle Y_1,Y_2,...,Y_n$ independent? Read this: Moment-generating function - Wikipedia, the free encyclopedia (Sum of independent random variables).
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