Named distributions

1. Raindrops are falling at an average rate of 20 per square inch per minute. You want to model the number of raindrops that fall per minute over an area that is five square inches.

   1. Is a Poisson distribution appropriate here? If so, what is the parameter \(\lambda\)?
   2. What is the probability that no rain drops fall in the area over a period of one minute?

2. Let’s say that the number of notifications you get in a day is a Poisson random variable with mean 20.

   1. What is the probability that you get exactly 20 messages?
   2. Given that you saw at least 20 notifications, write an expression for the probability that you got exactly 20.

3. Let \(X \sim \textrm{Poisson}(\lambda)\). Are the following Poisson variables? Why or why not?

1. \(X + X\)
2. \(X^2\)
3. \(X + 1\)

4. Remember the tennis problem? Two people are playing a tennis match, where the rule is whoever goes up by two first wins the match. Person A scores a point with probability \(p\) each round, the rounds are independent, and they play until someone wins.

   What is the expected number of rounds that will be played? Hint. Think of the first two games, the second two, and so on, as independent trials.

5. Hyperbolic cosine is a function defined by

\[\cosh x = \frac{e^x + e^{-x}}{2}.\]

1. Show that, if \(X \sim \textrm{Poisson}(\lambda)\), then

\[P(X \text{ is even}) = e^{-\lambda} \cosh \lambda.\]

2. Show that Poisson variables are more likely to be even than odd.

3. Say that \(\lambda = 4\). Given that \(X\) is even, what is the probability that it was \(0\)?

Expectations

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

\[ \begin{align*} p_X(-1) &= \frac{1}{2} \\ p_X(0) &= \frac{1}{3} \\ p_X(2) &= \frac{1}{6}. \end{align*} \]

1. What is the expectation of \(X\)?
2. What is the expectation of \(X^2\)?
3. What is the variance of \(X\)?
4. What is the standard deviation of \(X\)?

7. Let \(X \sim \text{Poisson}(\lambda)\).

   1. Compute \(E[X!]\), the average value of the factorial of \(X\).
   2. Compute \(E[2^X]\).

8. Let \(X \sim \text{Geometric}(p)\).

   1. For which values of \(p\) is \(E[2^X]\) finite?
   2. For which values of \(p\) is \(E[2^{-X}]\) finite?
   3. Compute the expectation \(E[e^{tX}]\) where \(t\) is a fixed parameter. (This is called the moment generating function of \(X\). We’ll come back to this in a few weeks!)

9. Let \(X \sim \textrm{Poisson}(\lambda)\). In class we’ll compute \(E[X]\) and \(E[X^2]\). The goal of this exercise is to compute \(E[X^3]\).

   1. Show that \(E[X f(X)] = \lambda E[f(X + 1)]\) for any function \(f\).
   2. Compute \(E[X^3]\) using the previous part with \(f(x) = x^2\).

Linearity of expectation and indicator variables

10. A fair coin is flipped \(n\) times. What is the expected number of times the pattern HTH will occur? Note that overlap is allowed: HTHTH would count as two occurrences of the pattern.

11. 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 ace is seen?

12. A group of \(n \geq 2\) people put their names on pieces of paper and drop them into a hat. What is the expected number of people that draw their own name? (Assume that no one has the same name among the \(n\) people.)