1. (a) Suppose deposit amounts, X, by customers to a savings account are uniformly distributed over the interval ($\theta$, $\beta$). Given also that the density of the distribution is defined by
$\qquad f(x) = \begin{cases} \frac{1}{\beta - \theta}, & \theta \le x \le \beta\\ 0, & \text{Otherwise} \end{cases}$
(i) Show that the function $f(x)$ is a probability density function.
(ii) Find the moment generating function of X.
(iii) Find the mean of X.
(iv) Obtain the variance of X.