91Ó°ÊÓ

Let \(X_{1}\) and \(X_{2}\) be independent geometric random variables having the same parameter \(p\). Guess the value of $$ P\left\\{X_{1}=i \mid X_{1}+X_{2}=n\right\\} $$ Hint: Suppose a coin having probability \(p\) of coming up heads is continually flipped. If the second head occurs on flip number \(n\), what is the conditional probability that the first head was on flip number \(i, i=1, \ldots, n-1 ?\) Verify your guess analytically.

The joint probability mass function of \(X\) and \(Y, p(x, y)\), is given by $$ \begin{array}{ll} p(1,1)=\frac{1}{9}, & p(2,1)=\frac{1}{3}, & p(3,1)=\frac{1}{9} \\ p(1,2)=\frac{1}{9}, & p(2,2)=0, & p(3,2)=\frac{1}{18} \\ p(1,3)=0, & p(2,3)=\frac{1}{6}, & p(3,3)=\frac{1}{9} \end{array} $$ Compute \(E[X \mid Y=i]\) for \(i=1,2,3\).

A coin having probability \(p\) of coming up heads is successively flipped until two of the most recent three flips are heads. Let \(N\) denote the number of flips. (Note that if the first two flips are heads, then \(N=2 .\) ) Find \(E[N]\).

Independent trials, resulting in one of the outcomes \(1,2,3\) with respective probabilities \(p_{1}, p_{2}, p_{3}, \sum_{i=1}^{3} p_{i}=1\), are performed. (a) Let \(N\) denote the number of trials needed until the initial outcome has occurred exactly 3 times. For instance, if the trial results are \(3,2,1,2,3,2,3\) then \(N=7\) Find \(E[N]\). (b) Find the expected number of trials needed until both outcome 1 and outcome 2 have occurred.

You have two opponents with whom you alternate play. Whenever you play \(A\), you win with probability \(p_{A}\); whenever you play \(B\), you win with probability \(p_{B}\), where \(p_{B}>p_{A}\). If your objective is to minimize the expected number of games you need to play to win two in a row, should you start with \(A\) or with \(B\) ? Hint: Let \(E\left[N_{i}\right]\) denote the mean number of games needed if you initially play \(i\). Derive an expression for \(E\left[N_{A}\right]\) that involves \(E\left[N_{B}\right] ;\) write down the equivalent expression for \(E\left[N_{B}\right]\) and then subtract.

A coin that comes up heads with probability \(p\) is continually flipped until the pattern \(\mathrm{T}, \mathrm{T}, \mathrm{H}\) appears. (That is, you stop flipping when the most recent flip lands heads, and the two immediately preceding it lands tails.) Let \(X\) denote the number of flips made, and find \(E[X]\).

Polya's urn model supposes that an urn initially contains \(r\) red and \(b\) blue balls. At each stage a ball is randomly selected from the urn and is then returned along with \(m\) other balls of the same color. Let \(X_{k}\) be the number of red balls drawn in the first \(k\) selections. (a) Find \(E\left[X_{1}\right]\) (b) Find \(E\left[X_{2}\right]\). (c) Find \(E\left[X_{3}\right]\). (d) Conjecture the value of \(E\left[X_{k}\right]\), and then verify your conjecture by a conditioning argument. (e) Give an intuitive proof for your conjecture. Hint: Number the initial \(r\) red and \(b\) blue balls, so the urn contains one type \(i\) red ball, for each \(i=1, \ldots, r ;\) as well as one type \(j\) blue ball, for each \(j=1, \ldots, b\). Now suppose that whenever a red ball is chosen it is returned along with \(m\) others of the same type, and similarly whenever a blue ball is chosen it is returned along with \(m\) others of the same type. Now, use a symmetry argument to determine the probability that any given selection is red.

Two players take turns shooting at a target, with each shot by player \(i\) hitting the target with probability \(p_{i}, i=1,2\). Shooting ends when two consecutive shots hit the target. Let \(\mu_{i}\) denote the mean number of shots taken when player \(i\) shoots first, \(i=1,2\) (a) Find \(\mu_{1}\) and \(\mu_{2}\). (b) Let \(h_{i}\) denote the mean number of times that the target is hit when player \(i\) shoots first, \(i=1,2\). Find \(h_{1}\) and \(h_{2}\).

Independent trials, each resulting in success with probability \(p\), are performed. (a) Find the expected number of trials needed for there to have been both at least \(n\) successes or at least \(m\) failures. Hint: Is it useful to know the result of the first \(n+m\) trials? (b) Find the expected number of trials needed for there to have been either at least \(n\) successes or at least \(m\) failures. Hint: Make use of the result from part (a).

A deck of \(n\) cards, numbered 1 through \(n\), is randomly shuffled so that all \(n !\) possible permutations are equally likely. The cards are then turned over one at a time until card number 1 appears. These upturned cards constitute the first cycle. We now determine (by looking at the upturned cards) the lowest numbered card that has not yet appeared, and we continue to turn the cards face up until that card appears. This new set of cards represents the second cycle. We again determine the lowest numbered of the remaining cards and turn the cards until it appears, and so on until all cards have been turned over. Let \(m_{n}\) denote the mean number of cycles. (a) Derive a recursive formula for \(m_{n}\) in terms of \(m_{k}, k=1, \ldots, n-1\). (b) Starting with \(m_{0}=0\), use the recursion to find \(m_{1}, m_{2}, m_{3}\), and \(m_{4}\). (c) Conjecture a general formula for \(m_{n}\). (d) Prove your formula by induction on \(n\). That is, show it is valid for \(n=1\), then assume it is true for any of the values \(1, \ldots, n-1\) and show that this implies it is true for \(n\). (e) Let \(X_{i}\) equal 1 if one of the cycles ends with card \(i\), and let it equal 0 otherwise, \(i=1, \ldots, n\). Express the number of cycles in terms of these \(X_{i}\). (f) Use the representation in part (e) to determine \(m_{n}\). (g) Are the random variables \(X_{1}, \ldots, X_{n}\) independent? Explain. (h) Find the variance of the number of cycles.