Let $X$ be a Unif$(-, )$ variable, that is, $X$ is Uniformly distributed over the interval $(-, )$.
Provide answers to the following to two decimal places.

Part a) Find the MGF of $X$, evaluated at the point $t =$.


Part b) Let $X_1, X_2, ..., X_n$ be independent Unif$(-, )$ variables. Let \begin{align*} Y = \sum_{i=1}^n X_i. \end{align*} Find the MGF $M_Y(t)$ of $Y$. Evaluate the MGF at the point $t =$ in the case $n =$.


Part c) Find the standard deviation of $Y$.


Part d) Standardize $Y$ to create a new variable $Z$ with mean zero and standard deviation 1. Find the MGF $M_{Z}\left( t\right)$ of $Z.$ Evaluate $M_{Z}\left( t\right)$ at the point $t =$ in the case $n =$.


Part e) Suppose we let $n$ tend to infinity. Find $\lim_{n \to \infty} M_Z(t)$, and evaluate it at the point $t =$. Do you recognise the MGF?