91Ó°ÊÓ

A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die. Determine her expected winnings.

The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is randomly chosen and the object of the game is to guess the chosen three.(a) How many solutions are possible? In one version of the game, the selection is made and then each of the players is randomly given three of the remaining cards. Let \(S, W\) and \(R\) be, respectively, the numbers of suspects, weapons, and rooms in the set of three cards given to a specified player. Also, let \(X\) denote the number of solutions that are possible after that player observes his or her three cards. (b) Express \(X\) in terms of \(S, W,\) and \(R\) (c) Find \(E[X]\)

\(N\) people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the \(\left(\begin{array}{l}N \\ 2\end{array}\right)\)pairs of people is, independently, a pair of friends with probability \(p,\) find the expected number of occupied tables. Hint: Let \(X_{i}\) equal 1 or \(0,\) depending on whether the \(i\) th arrival sits at a previously unoccupied table.

Consider 3 trials, each having the same probability of success. Let \(X\) denote the total number of successes in these trials. If \(E[X]=1.8\) what is (a) the largest possible value of \(P\\{X=3\\} ?\) (b) the smallest possible value of \(P\\{X=3\\} ?\) In both cases, construct a probability scenario that results in \(P\\{X=3\\}\) having the stated value. Hint: For part (b), you might start by letting \(U\) be a uniform random variable on (0,1) and then defining the trials in terms of the value of \(U\)

Cards from an ordinary deck of 52 playing cards are turned face up one at a time. If the 1 st card is an ace, or the 2 nd a deuce, or the 3 rd a three, or \(\dots,\) or the 13 th a king, or the 14 an ace, and so on, we say that a match occurs. Note that we do not require that the \((13 n+1)\) th card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.

There are 4 different types of coupons, the first 2 of which compose one group and the second 2 another group. Each new coupon obtained is type \(i\) with probability \(p_{i},\) where \(p_{1}=p_{2}=1 / 8, p_{3}=\) \(p_{4}=3 / 8 .\) Find the expected number of coupons that one must obtain to have at least one of (a) all 4 types; (b) all the types of the first group; (c) all the types of the second group; (d) all the types of either group.

The joint density function of \(X\) and \(Y\) is given by $$ f(x, y)=\frac{1}{y} e^{-(y+x / y)}, \quad x>0, y>0 $$ Find \(E[X], E[Y],\) and show that \(\operatorname{Cov}(X, Y)=1\)

Consider a graph having \(n\) vertices labeled \(1,2, \ldots, n,\) and suppose that, between each of the \(\left(\begin{array}{l}n \\ 2\end{array}\right)\) pairs of distinct vertices, an edge is independently present with probability \(p .\) The degree of vertex \(i,\) designated as \(D_{i}\), is the number of edges that have vertex \(i\) as one of their vertices. (a) What is the distribution of \(D_{i} ?\) (b) Find \(\rho\left(D_{i}, D_{j}\right),\) the correlation between \(D_{i}\) and \(D_{j}\)

Let \(X\) be the value of the first die and \(Y\) the sum of the values when two dice are rolled. Compute the joint moment generating function of \(X\) and \(Y\)

Successive weekly sales, in units of one thousand dollars, have a bivariate normal distribution with common mean \(40,\) common standard deviation 6 and correlation .6. (a) Find the probability that the total of the next 2 weeks' sales exceeds \(90 .\) (b) If the correlation were .2 rather than \(.6,\) do you think that this would increase or decrease the answer to (a)? Explain your reasoning. (c) Repeat (a) when the correlation is .2.