Let \(Z\sim\text{N}(0,\,1)\), and recall what that means:

\[ \begin{aligned} f_Z(z)&=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}z^2\right), && -\infty<z<\infty\\ M_Z(t)&=\exp\left(\frac{1}{2}t^2\right), && -\infty<t<\infty. \end{aligned} \]

Next, let \(X=\mu+\sigma Z\) for some constants \(\mu\in\mathbb{R}\) and \(\sigma>0\).

1. Use the change-of-variables formula to derive density of \(X\). What is its distribution?
2. What is the moment-generating function of \(X\)?
3. What are the mean and variance of \(X\)? Make sure you justify your answer with some type of derivation.
4. Consider \(X_1,\,X_2,\,...,\,X_n\overset{\text{iid}}{\sim}\text{N}(\mu,\,\sigma^2)\) and derive the distribution of their sum and their average.