Let $X_1,\dots,X_n$ be independent random variables with a normal distribution having mean $1$ and variance $2$. Find the moment generating function for $n^{-1/2}(S_n-n)$.

Umm. Our book doesn't have any examples on finding these mgfs. Sorry for spamming mgf questions lately, I kinda suck at this and our teacher did an awesome job delaying teaching this subject until there was almost no time before the end of lectures :( and the book does not cover it very well, even though the problems are messy.

Here are the tips

  • You know the formula for the MGF of the normal distribution.
  • You know that the MGF of a sum of i.i.d variables is the product of their MGFs.
  • You know that if $Y$ is normal, then $a Y$ is normal for any scalar $a$.

The second of these points is immediate once you write down the formula for the MGF of $S_n$.

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