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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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Consider a communication source that transmits packets containing digitized speech. After each transmission, the receiver sends a message indicating whether the transmission was successful or unsuccessful. If a transmission is unsuccessful, the packet is re-sent. Suppose a voice packet can be transmitted a maximum of \({\rm{10}}\) times. Assuming that the results of successive transmissions are independent of one another and that the probability of any particular transmission being successful is \({\rm{p}}\), determine the probability mass function of the rv \({\rm{X = }}\)the number of times a packet is transmitted. Then obtain an expression for the expected number of times a packet is transmitted.
Twenty pairs of individuals playing in a bridge tournament have been seeded \({\rm{1,}}....{\rm{,20}}\). In the first part of the tournament, the \({\rm{20}}\) are randomly divided into 10 east鈥� west pairs and \({\rm{10}}\) north鈥搒outh pairs. a. What is the probability that x of the top \({\rm{10}}\) pairs end up playing east鈥搘est? b. What is the probability that all of the top five pairs end up playing the same direction? c. If there are \({\rm{2}}\)n pairs, what is the pmf of X = the number among the top n pairs who end up playing east鈥搘est? What are E(X) and V(X)?
A reservation service employs five information operators who receive requests for information independently of one another, each according to a Poisson process with rate a \({\rm{\alpha = 2}}\) per minute.
a. What is the probability that during a given \({\rm{1 - min}}\) period, the first operator receives no requests?
b. What is the probability that during a given \({\rm{1 - min}}\) period, exactly four of the five operators receive no requests?
c. Write an expression for the probability that during a given \({\rm{1 - min}}\) period, all of the operators receive exactly the same number of requests.
A second-stage smog alert has been called in a certain area of Los Angeles County in which there are \({\rm{50}}\) industrial firms. An inspector will visit \({\rm{10}}\) randomly selected firms to check for violations of regulations. a. If \({\rm{15}}\) of the firms are actually violating at least one regulation, what is the pmf of the number of firms visited by the inspector that are in violation of at least one regulation? b. If there are \({\rm{500}}\) firms in the area, of which \({\rm{150}}\) are in violation, approximate the pmf of part (a) by a simpler pmf. c. For X = the number among the 10 visited that are in violation, compute E(X) and V(X) both for the exact pmf and the approximating pmf in part (b).
a. Show that b(x; n,\({\rm{1 - }}\)p) = b(n - x; n, p). b. Show that B(x; n,\({\rm{1 - }}\)p) =\({\rm{1 - }}\)B(n - x -\({\rm{1}}\); n, p). (Hint: At most x S鈥檚 is equivalent to at least (n - x) F鈥檚.) c. What do parts (a) and (b) imply about the necessity of including values of p greater than\({\rm{.5}}\)in Appendix Table A\({\rm{.1}}\)?
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