Roughly 60 percent of the T-cells of a healthy person are of the CD4 type, whereas the percentage of the T-cells that are of CD4 type appears to decrease continually in AIDS sufferers. For each value of $i, i=2, \ldots, 12,$ find your expected return if you employ the strategy of stopping the first time that a value at least as large as $i$ appears. The first door leads to a tunnel that returns him to his cell after 2 days' travel.

Expected value Consider a random variable Y = r(X) for some function r, e.g. Y = X2 + 3 so in this case r(x) = x2 + 3. Properties of expectation Linearity. Hint: Let $X_{i}$ equal 1 or $0,$ depending on whether the $i$ th arrival sits at a previously unoccupied table.A total of $n$ balls, numbered 1 through $n,$ are put into $n$ urns, also numbered 1 through $n$ in such a way that ball $i$ is equally likely to go into any of the urns $1,2, \ldots, i .$ Find,Consider 3 trials, each having the same probability of success. If the coin lands heads, then she wins twice, and if tails, then one-half of the value that appears on the die.

Properties of Expectation Educators. NOTE: The preceding has possible applications to understanding the AIDS disease. 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. The bank then rolls the dice to determine the outcome according to the following rule: Player $i, i=1,2,$ wins if his roll is strictly greater than the bank's. Let $A$ denote a specificd one of the players, and let $X$ denote the amount that is received by $A$(a) Compute the expected total prize shared by the players. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the $p$ -value of the coin can be regarded as being the value of a random variable that is uniformly distributed over $[0,1] .$ If a coin is selected at random from the urn and flipped twice, compute the probability that,In Problem $70,$ suppose that the coin is tossed $n$ times. Let X 1 and X 2 be two random variables and c 1;c 2 be two real numbers, then E[c 1X 1 + c 2X 2] = c 1EX 1 + c 2EX 2: Taking these two properties, we say that expectation is a positive linear functional. (a) How many solutions are possible? Determine her expected winnings.The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. If a small pill is chosen, then that pill is eaten. For $i=1,2,$ let,Consider a graph having $n$ vertices labeled $1,2, \ldots, n,$ and suppose that, between each of the,A fair die is successively rolled. Compute the expected number of cards that need to be turned face up in order to obtain,Let $X$ be the number of 1 's and $Y$ the number of 2 's that occur in $n$ rolls of a fair die. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.Let $X_{1}, X_{2}, \ldots$ be a sequence of independent and identically distributed continuous random variables. If 20 fish are caught, what are the mean and variance of the number of carp among the $20 ?$ What assumptions are you making?A group of 20 people consisting of 10 men and,Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent random variables having an unknown continuous distribution function $F,$ and let $Y_{1}, Y_{2}, \ldots, Y_{m}$ be independent random variables having an unknown continuous distribution function $G .$ Now order those $n+m$ variables, and let,Between two distinct methods for manufacturing certain goods, the quality of goods produced by method $i$ is a continuous random variable having distribution $F_{i}, i=1,2 .$ Suppose that $n$ goods are produced by method 1 and $m$ by method $2 .$ Rank the $n+m$ goods according to quality, and let,If $X_{1}, X_{2}, X_{3},$ and $X_{4}$ are (pairwise) uncorrelated random variables, each having mean 0 and variance $1,$ compute the correlations of,Consider the following dice game, as played at a certain gambling casino: Players 1 and 2 roll a pair of dice in turn.