91Ó°ÊÓ

\begin{aligned} &\text { (a) Show that }\\\ &\operatorname{Cov}(X, Y)=\operatorname{Cov}(X, E[Y \mid X]) \end{aligned} (b) Suppose, that, for constants \(a\) and \(b\), $$ E[Y \mid X]=a+b X $$ Show that $$ b=\operatorname{Cov}(X, Y) / \operatorname{Var}(X) $$

Suppose \(X\) is a Poisson random variable with mean \(\lambda .\) The parameter \(\lambda\) is itself a random variable whose distribution is exponential with mean \(1 .\) Show that \(P[X=\) \(n\\}=\left(\frac{1}{2}\right)^{n+1}\)

Two players alternate flipping a coin that comes up heads with probability \(p\). The first one to obtain a head is declared the winner. We are interested in the probability that the first player to flip is the winner. Before determining this probability, which we will call \(f(p)\), answer the following questions. (a) Do you think that \(f(p)\) is a monotone function of \(p ?\) If so, is it increasing or decreasing? (b) What do you think is the value of \(\lim _{p \rightarrow 1} f(p) ?\) (c) What do you think is the value of \(\lim _{p \rightarrow 0} f(p) ?\) (d) Find \(f(p)\).

\(A\) and \(B\) roll a pair of dice in turn, with \(A\) rolling first. A's objective is to obtain a sum of 6 , and \(B\) 's is to obtain a sum of 7 . The game ends when either player reaches his or her objective, and that player is declared the winner. (a) Find the probability that \(A\) is the winner. (b) Find the expected number of rolls of the dice. (c) Find the variance of the number of rolls of the dice.

Consider a gambler who on each bet either wins 1 with probability \(18 / 38\) or loses 1 with probability \(20 / 38\). (These are the probabilities if the bet is that a roulette wheel will land on a specified color.) The gambler will quit either when he or she is winning a total of 5 or after 100 plays. What is the probability he or she plays exactly 15 times? Sh

An urn contains \(n\) white and \(m\) black balls that are removed one at a time. If \(n>m\), show that the probability that there are always more white than black balls in the urn (until, of course, the urn is empty) equals ( \(n-m) /(n+m)\). Explain why this probability is equal to the probability that the set of withdrawn balls always contains more white than black balls. (This latter probability is \((n-m) /(n+m)\) by the ballot problem.)

A coin that comes up heads with probability \(p\) is flipped \(n\) consecutive times. What is the probability that starting with the first flip there are always more heads than tails that have appeared?

Consider a large population of families, and suppose that the number of children in the different families are independent Poisson random variables with mean \(\lambda\). Show that the number of siblings of a randomly chosen child is also Poisson distributed with mean \(\lambda\).

Use the conditional variance formula to find the variance of a geometric random variable.