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Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y, and state which Y value is associated with each one.

Let X have a Poisson distribution with parameter \({\rm{\mu }}\). Show that E(X) =\({\rm{\mu }}\) directly from the definition of expected value. (Hint: The first term in the sum equals \({\rm{0}}\), and then x can be cancelled. Now factor out \({\rm{\mu }}\) and show that what is left sums to \({\rm{1}}\).)

Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter\({\rm{\alpha }}\), the expected number of trees per acre, equal to\({\rm{80}}\). a. What is the probability that in a certain quarter-acre plot, there will be at most\({\rm{16}}\)trees? b. If the forest covers\({\rm{85,000}}\)acres, what is the expected number of trees in the forest? c. Suppose you select a point in the forest and construct a circle of radius\({\rm{.1}}\)mile. Let X = the number of trees within that circular region. What is the pmf of X? (Hint:\({\rm{1}}\)sq mile\({\rm{ = 640}}\)acres.)

Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate \(\alpha = 10\)per hour. Suppose that with probability \(.{\bf{5}}\) an arriving vehicle will have no equipment violations. a. What is the probability that exactly ten arrive during the hour and all ten have no violations? b. For any fixed \(y \ge 10\), what is the probability that y arrives during the hour, of which ten have no violations? c. What is the probability that ten 鈥渘o-violation鈥� cars arrive during the next hour?

Consider a deck consisting of seven cards, marked\({\rm{1,2, \ldots }}\),\({\rm{7}}\). Three of these cards are selected at random. Define an rv \({\rm{W}}\) by \({\rm{W = }}\) the sum of the resulting numbers, and compute the pmf of \({\rm{W}}\). Then compute \({\rm{\mu }}\) and\({{\rm{\sigma }}^{\rm{2}}}\). (Hint: Consider outcomes as unordered, so that \({\rm{(1,3,7)}}\) and \({\rm{(3,1,7)}}\) are not different outcomes. Then there are \({\rm{35}}\) outcomes, and they can be listed. (This type of rv actually arises in connection with a statistical procedure called Wilcoxon's rank-sum test, in which there is an \({\rm{x}}\) sample and a \({\rm{y}}\) sample and \({\rm{W}}\)is the sum of the ranks of the \({\rm{x}}\)'s in the combined sample)

Of all customers purchasing automatic garage-door openers, 75% purchase a chain-driven model. Let \({\bf{X}}{\rm{ }}{\bf{5}}\) the number among the next \({\bf{15}}\) purchasers who select the chain-driven model.

a. What is the pmf of \({\bf{X}}\)?

b. Compute \({\bf{P}}\left( {{\bf{X}}{\rm{ }}.{\rm{ }}{\bf{10}}} \right).\)

c. Compute \({\bf{P}}\left( {{\bf{6}}{\rm{ }}\# {\rm{ }}{\bf{X}}{\rm{ }}\# {\rm{ }}{\bf{10}}} \right).\)

d. Compute \({\bf{m}}\) and s2 .

e. If the store currently has in stock \({\bf{10}}\) chain-driven models and \({\bf{8}}\) shaft-driven models, what is the probability that the requests of these 15 customers can all be met from existing stock?

A k-out-of-n system is one that will function if and only if at least k of the n individual components in the system function. If individual components function independently of one another, each with probability.\(9\), what is the probability that a 3-out-of-5 system functions?

An individual named Claudius is located at the point 0 in the accompanying diagram. Using an appropriate randomization device (such as a
tetrahedral die, one having four sides), Claudius first moves to one of the four locations B1, B2, B3, B4. Once at one of these locations, another randomization device is used to decide whether Claudius next returns to 0 or next visits one of the other two adjacent points. This process then continues; after each move, another move to one of the (new) adjacent points is determined by tossing an appropriate die or coin.

a. Let X = the number of moves that Claudius makes before first returning to 0. What are possible values of X? Is X discrete or continuous?

b. If moves are allowed also along the diagonal paths connecting 0 to A1, A2, A3, and A4, respectively, answer the questions in part (a).