Problems: 7.75

Theoretical Exercises: 7.46, 7.47, 7.49, 7.51

Also:

1. Find the moment generating function for a geometric random variable. Check your answer with the table on page 339.
2. Find the moment generating function for a negative binomial random variable. Check your answer with the table on page 339. Hint: Make use of the moment generating function of a geometric random variable.
3. Let $X$ be a Poisson random variable with mean $\lambda$. Use the moment generating function of $X$ to find the mean and variance of $X$.
4. Suppose $X$ has moment generating function $M(t) = \frac{1}{2}(e^{t} + e^{-t})$. Find the mean and variance of $X$.