Exercises 6.10: 18, 21

Also:

1. Suppose $X$ is uniform on $(a, b)$. Show that the moment generating function of $X$ is $M_X(t) = \frac{e^{tb} - e^{ta}}{t(b - a)}$.
2. Suppose $X$ is geometric with probability of success $p$ and let $q = 1 - p$. Show that the moment generating function of $X$ is $M_X(t) = \frac{p}{1 - qe^t}$ for $qe^t < 1$.
3. Suppose $X$ has moment generating function $M_X(t) = \frac{1}{(1 - 2t)^3}$ for $t < \frac{1}{2}$. Find the mean and variance of $X$.
4. Suppose $X \sim \text{Expo}(\lambda)$. Use the moment generating function of $X$ to show that the $k$th moment of $X$ is $\mu_k = \frac{k!}{\lambda^k}$.

For Problem Set due 2 November: 18