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Chapter 3: Q5E (page 98)
If the sample space S is an infinite set, does this necessarily imply that any rv X defined from S will have an infinite set of possible values? If yes, say why. If no, give an example.
No. Consider the random variable X defined as 鈥楪etting a Head鈥� in a random experiment of flipping a coin n times.
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A plan for an executive travellers鈥� club has been developed by an airline on the premise that \({\rm{10\% }}\) of its current customers would qualify for membership.
a. Assuming the validity of this premise, among \({\rm{25}}\) randomly selected current customers, what is the probability that between \({\rm{2}}\) and \({\rm{6}}\) (inclusive) qualify for membership?
b. Again assuming the validity of the premise, what are the expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of \({\rm{100}}\) current customers?
c. Let \({\rm{X}}\) denote the number in a random sample of \({\rm{25}}\) current customers who qualify for membership. Consider rejecting the company鈥檚 premise in favour of the claim that \({\rm{p > 10}}\) if \({\rm{x}} \ge {\rm{7}}\). What is the probability that the company鈥檚 premise is rejected when it is actually valid?
d. Refer to the decision rule introduced in part (c). What is the probability that the company鈥檚 premise is not rejected even though \({\rm{p = }}{\rm{.20}}\) (i.e., \({\rm{20\% }}\) qualify)?
A personnel director interviewing \({\rm{11}}\) senior engineers for four job openings has scheduled six interviews for the first day and five for the second day of interviewing. Assume that the candidates are interviewed in random order. a. What is the probability that x of the top four candidates are interviewed on the first day? b. How many of the top four candidates can be expected to be interviewed on the first day?
Eighteen individuals are scheduled to take a driving test at a particular DMV office on a certain day, eight of whom will be taking the test for the first time. Suppose that six of these individuals are randomly assigned to a particular examiner, and let X be the number among the six who are taking the test for the first time. a. What kind of a distribution does X have (name and values of all parameters)? b. Compute P(X\({\rm{ = 2}}\)), P(X\( \le {\rm{ 2}}\)), and P(X\( \ge {\rm{ 2}}\)). c. Calculate the mean value and standard deviation of X.
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)
The article 'should You Report That Fender Bender?" (Consumer Reports, Sept. 2013: 15) reported that 7 in 10 auto accidents involve a single vehicle (the article recommended always reporting to the insurance company an accident involving multiple vehicles). Suppose 15 accidents are randomly selected. Use Appendix Table A.l to answer each of the following questions.
a. What is the probability that at most 4 involve a single vehicle?
b. What is the probability that exactly 4 involve a single vehicle?
c. What is the probability that exactly 6 involve multiple vehicles?
d. What is the probability that between 2 and 4 , inclusive, involve a single vehicle?
e. What is the probability that at least 2 involve a single vehicle?
f. What is the probability that exactly 4 involve a single vehicle and the other 11 involve multiple vehicles?
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