Probability problems day 9

1. Let \(X\) be a random variable with pmf

\[p_X(-1) = \frac{1}{2}\] \[p_X(c) = \frac{1}{2}\]

where \(c > 0\) is a constant.

1. What is \(E[X]\)?
2. What is \(E[X^2]\)?
3. What is \(V(X)\)?
4. What is \(SD(X)\)?

2. Let \(X \sim \text{Poisson}(\lambda)\). Compute \(E[(X + 1)^{-1}]\). Does this equal \(1 / E[X + 1]\)?

3. Shuffle a standard deck of 52 cards and deal them one at a time. What is the expected number of cards drawn before the first king is seen?

4. If \(X \sim \text{Poisson}(\lambda)\), is \(X + X\) a Poisson variable?