Let $X_1,X_2,\dots,X_n$ be a random sample from a Standard Norm Dist with MGF $M_X(t)=e^{0.5t^2}$. Let $\overline{X}=\frac{1}{n}(X_1+X_2 +\dots + X_n)$. Determine the MGF of $\overline{X}$.
I managed to get the MGF of $\overline{X}=[M_X(t/n)]^n$ by using the expectation of $e^{t \overline{X}}$. How do I proceed?
Thanks.
Recall that the moment-generating function of a sum of independent random variables $X_1,\ldots,X_n$ is the product of the moment-generating functions of each of the $X_i$. Also, $\mathrm{Var}(cX)=c^2\mathrm{Var}(X)$ for any real number $c$. Hence $$\begin{align*} \mathbb E\left[e^{t\overline X}\right] &= \mathbb E\left[e^{t\frac1n\sum_{i=1}^n X_i}\right]\\ &= \mathbb E\left[\prod_{i=1}^n e^{t\frac1n X_i} \right]\\ &= \prod_{i=1}^n\mathbb E\left[ e^{t\frac1n X_i} \right]\\ &= \mathbb E\left[e^{t\frac1n X_1} \right]^n\\ &= \left(e^{\frac12\cdot \frac1{n^2}t} \right)^n\\ &= e^{\frac12\cdot\frac1n t}. \end{align*}$$ It follows that $$\overline X \sim\mathcal N\left(0,\frac1n\right) .$$