91Ó°ÊÓ

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.

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table.

a. What is the probability that the flight will accommodateall ticketed passengers who show up?

b. What is the probability that not all ticketed passengerswho show up can be accommodated?

c. If you are the first person on the standby list (whichmeans you will be the first one to get on the plane ifthere are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table.

Calculate the probability of each of the following events.

a. {at most three lines are in use}

b. {fewer than three lines are in use}

c. {at least three lines are in use}

d. {between two and five lines, inclusive, are in use}

e. {between two and four lines, inclusive, are not in use}

f. {at least four lines are not in use}

A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let Y = the number of forms required of the next applicant. The probability that y forms are required is known to be proportional to y—that is,\(p\left( y \right) = ky\)for\(y = 1, \ldots ,5\).

a. What is the value of k? (Hint:\(\sum\limits_{y = 1}^5 {p\left( y \right)} = 1\))

b. What is the probability that at most three forms arerequired?

c. What is the probability that between two and fourforms (inclusive) are required?

d. Could \(p\left( y \right) = \frac{{{y^2}}}{{50}}\)for \(y = 1, \ldots ,5\)be the pmf of Y?

Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives circuit boards in batches of five. Two boards are selected from each batch for inspection. We can represent possible outcomes

of the selection process by pairs. For example, the pair (1, 2) represents the selection of boards 1 and 2 for inspection.

a. List the ten different possible outcomes.

b. Suppose that boards 1 and 2 are the only defective boards in a batch. Two boards are to be chosen at random. Define X to be the number of defective boards observed among those inspected. Find the probability distribution of X.

c. Let F(x) denote the cdf of X. First determine \(F\left( 0 \right) = P\left( {X \le 0} \right)\), F(1), and F(2); then obtain F(x) for allother x.

Some parts of California are particularly earthquake prone. Suppose that in one metropolitan area, 25% of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let X

denote the number among the four who have earthquake insurance.

a. Find the probability distribution of X. (Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS,

with probability (.25)(.75)(.25)(.25) and associated X value 3. There are 15 other outcomes.)

b. Draw the corresponding probability histogram.

c. What is the most likely value for X?

d. What is the probability that at least two of the four selected have earthquake insurance?

A new battery’s voltage may be acceptable (A) or unacceptable (U). A certain flashlight requires two batteries, so batteries will be independently selected and tested until two acceptable ones have been found. Suppose that 90% of all batteries have acceptable voltages. Let Y denote the number of batteries that must be tested.

a. What is\(p\left( 2 \right)\), that is, \(P\left( {Y = 2} \right)\),?

b. What is\(p\left( 2 \right)\)? (Hint: There are two different outcomes that result in\(Y = 3\).)

c. To have \(Y = 5\), what must be true of the fifth battery selected? List the four outcomes for which Y = 5 and then determine\(p\left( 5 \right)\).

d. Use the pattern in your answers for parts (a)–(c) to obtain a general formula \(p\left( y \right)\).

Two fair six-sided dice are tossed independently. Let M = the maximum of the two tosses (so M(1,5) =5, M(3,3) = 3, etc.).

a. What is the pmf of M? (Hint: First determine p(1), then p(2), and so on.)

b. Determine the cdf of M and graph it.

A library subscribes to two different weekly news magazines, each of which is supposed to arrive in Wednesday’s mail. In actuality, each one may arrive on Wednesday, Thursday, Friday, or Saturday. Suppose the two arrive independently of one another, and for each one\(P\left( {Wed.} \right) = 0.3\), \(P\left( {Thurs.} \right) = 0.4\), \(P\left( {Fri.} \right) = 0.2\), and\(P\left( {Sat.} \right) = 0.1\). Let Y = the number of days beyond Wednesday that it takes for both magazines to arrive (so possible Y values are 0, 1, 2, or 3). Compute the pmf of Y. (Hint: There are 16 possible outcomes; \(Y\left( {W,W} \right) = {\bf{0}}\),\(Y\left( {F,Th} \right) = 2\), and so on.)

Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number ofpeople who arrive late for the seminar.

a. Determine the probability mass function of X. (Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.)

b. Obtain the cumulative distribution function of X, and use it to calculate\(P\left( {2 \le X \le 6} \right)\).