mgfs

1. A fair die is rolled twice. Call \(X\) the first roll and \(Y\) the second roll.

1. Find the mgf of \(X\) (and therefore \(Y\)).
2. Find the mgf of \(X + Y\).
3. Use the mgf of \(X + Y\) to find the variance of \(X + Y\).

2. Let \(X, Y \sim \text{Poisson}(\lambda)\).

1. Explain why \(2X\) is not a Poisson variable.
2. Compute the mgf of \(2X + Y\). Is this a Poisson variable?

3. Let \(\Phi(x)\) be the standard normal cdf.

1. Find an expression for \(E[\Phi(Z) e^Z]\) as an unevaluated integral.
2. Evaluate \(E[e^Z]\) (easy!) and \(E[\Phi(Z)]\) (hard!).

4. Let \(X\) be a random variable with density

\[ f(x) = \begin{cases} \frac{1}{2}\sin x & 0 \leq x \leq \pi \\ 0 & \text{otherwise} \end{cases} \]

1. Find the moment generating function of \(X\).
2. Find the second moment of \(X\).

5. Let \(X\) and \(Y\) be independent Geometric(\(p\)) variables. Find the mgf of \(X + Y\). What is the distribution of \(X + Y\)?

Joint distributions

6. A fair coin is flipped twice. Let \(X\) be the number of Heads in the two tosses, and \(Y\) be the indicator variable for the tosses landing the same way.

1. Find the joint PMF of \(X\) and \(Y\).
2. Find the marginal PMFs of \(X\) and \(Y\).
3. Are \(X\) and \(Y\) independent?

6. One of two doctors, Dr. Hibbert and Dr. Nick, is called upon to perform a series of \(n\) surgeries. (This means that one doctor is chosen at random, and then does all \(n\) surgeries.) Let \(H\) be the indicator variable for Dr. Hibbert performing the surgeries, and suppose that \(E(H) = p\). Given that Dr. Hibbert is performing the surgeries, each surgery is successful with probability \(a\), independently. Given that Dr. Nick is performing the surgeries, each surgery is successful with probability \(b\), independently. Let \(X\) be the number of successful surgeries.

1. Find the joint PMF of \(H\) and \(X\).
2. Find the marginal PMF of \(X\).
3. Find the conditional PMF of H given \(X = k\).