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introduction to probability statistics
Introduction To Probability Models 6th Edition Sheldon M. Ross - Solutions
A small barbershop, operated by a single barber, has room for at most two customers. Potential customers arrive at a Poisson rate of three per hour, and the successive service times are independent exponential random variables with mean hour. What is (a) the average number of customers in the shop?
Potential customers arrive at a full-service, one-pump gas station at a Poisson rate of 20 cars per hour. However, customers will only enter the station for gas if there are no more than two cars (including the one currently being attended to) at the pump. Suppose the amount of time required to
A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at most three customers, (a) what fraction of potential customers enter the system? (b) what would the
The following problem arises in molecular biology. The surface of a bacterium is supposed to consist of several sites at which foreign molecules-some acceptable and some not-become attached. We consider a particular site and assume that molecules arrive at the site according to a Poisson process
Each time a machine is repaired it remains up for an exponentially distributed time with rate 2.It then fails, and its failure is either of two types. If it is a type 1 failure, then the time to repair the machine is exponential with rate ; if it is a type 2 failure, then the repair time is
After being repaired, a machine functions for an exponential time with rate and then fails. Upon failure, a repair process begins. The repair process proceeds sequentially through k distinct phases. First a phase 1 repair must be performed, then a phase 2, and so on. The times to complete these
A single repairperson looks after both machines 1 and 2.Each time it is repaired, machine i stays up for an exponential time with rate A,i = 1,2. When machine fails, it requires an exponentially distributed amount of work with rate , to complete its repair. The repairperson will always service
There are two machines, one of which is used as spare. A working machine will function for an exponential time with rate and will then fail. Upon failure, it is immediately replaced by the other machine if that one is in working order, and it goes to the repair facility. The repair facility
Suppose that when both machines are down in Exercise 20 a second repairperson is called in to work on the newly failed one. Suppose all repair times remain exponential with rate . Now find the proportion of time at least one machine is working, and compare your answer with the one obtained in
Customers arrive at a single server queue in accordance with a Poisson process having rate A. However, an arrival that finds n customers already in the system will only join the system with probability 1/(n + 1). That is, with probability n/(n + 1) such an arrival will not join the system. Show
A job shop consists of three machines and two repairmen. The amount of time a machine works before breaking down is exponentially distributed with mean 10.If the amount of time it takes a single repairman to fix a machine is exponentially distributed with mean 8, then (a) what is the average number
Consider a taxi station where taxis and customers arrive in accordance with Poisson processes with respective rates of one and two per minute. A taxi will wait no matter how many other taxis are present. However, if an arriving customer does not find a taxi waiting, he leaves. Find (a) the average
Customers arrive at a service station, manned by a single server who serves at an exponential rate , at a Poisson rate A. After completion of service the customer then joins a second system where the server serves at an exponential rate 2.Such a system is called a tandem or sequential queueing
If (X()] and [Y(t)) are independent continuous-time Markov chains, both of which are time reversible, show that the process (X(t), Y(1)) is also a time reversible Markov chain.
Consider a set of n machines and a single repair facility to service these machines. Suppose that when machine i, i = 1,..., n, fails it requires an exponentially distributed amount of work with rate , to repair it. The repair facility divides its efforts equally among all failed machines in the
Consider a graph with nodes 1, 2,...,n and the arcs (i,j), ij,,,1,, n. (See Section 3.6.2 for appropriate definitions.) Suppose that a particle moves along this graph as follows: Events occur along the arcs (i,j) according to independent Poisson processes with rates 2.An event along arc (i,j)
A total of N customers move about among 7 servers in the following manner. When a customer is served by server i, he then goes over to server j.ji, with probability 1/(r - 1). If the server he goes to is free, then the customer enters service; otherwise he joins the queue. The service times are all
Customers arrive at a two-server station in accordance with a Poisson process having rate 2.Upon arriving, they join a single queue. Whenever a server completes a service, the person first in line enters service. The service times of server i are exponential with rate , 1, 2, where > . An arrival
Consider two M/M/1 queues with respective parameters i 1, 2.Suppose they share a common waiting room that can hold at most 3 customers. That is, whenever an arrival finds his server busy and 3 customers in the waiting room, then he goes away. Find the limiting probability that there will be n queue
Four workers share an office that contains four telephones. At any time, each worker is either "working" or "on the phone." Each "working" period of worker i lasts for an exponentially distributed time with rate A,, and each "on the phone" period lasts for an exponentially distributed time with
Consider a time reversible continuous-time Markov chain having infinitesimal transition rates q, and limiting probabilities (P). Let A denote a set of states for this chain, and consider a new continuous-time Markov chain with transition rates q given by cau ifie A, je A otherwise where c is an
Consider a system of n components such that the working times of component i, i = 1,..., n, are exponentially distributed with rate ,. When failed, however, the repair rate of component i depends on how many other components are down. Specifically, suppose that the instantaneous repair rate of
For the continuous-time Markov chain of Exercise 3 present a uniformized version.
In Example 6.20, we computed m(t) =E[O(t)], the expected occupa- tion time in state 0 by time for the two-state continuous-time Markov chain starting in state 0.Another way of obtaining this quantity is by deriving a differential equation for it. (a) Show that m(t + h) = m(t) + Poo(!)h + o(h) (b)
Let O(t) be the occupation time for state 0 in the two-state continuous- time Markov chain. Find E[O(t) | X(0) = 1].
Consider the two-state continuous-time Markov chain. Starting in state 0, find Cov[X(s), X(t)].
Let Y denote an exponential random variable with rate that is independent of the continuous-time Markov chain (X()) and let P = P(X(Y) =j|X(0) = i} (a) Show that 1 Py = v; +2 2 U; +2 where 5 is 1 when i j and 0 when i j. (b) Show that the solution of the preceding set of equations is given by P =
(a) Show that Approximation 1 of Section 6.8 is equivalent to uniformizing the continuous-time Markov chain with a value v such that vt n and then approximating P,(t) by P*". (b) Explain why the preceding should make a good approximation. Hint: What is the standard deviation of a Poisson random
A box contains three marbles: one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box then replacing it in the box and drawing a second marble from the box. What is the sample space? If, at all times, each marble in the box is equally likely to be
Repeat 1 when the second marble is drawn without replacing the first marble.
A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, then what is the probability that it will be tossed exactly four times?
Let E, F, G be three events. Find expressions for the events that of E, F, G (a) only F occurs, (b) both E and F but not G occurs, (c) at least one event occurs, (d) at least two events occur, (e) all three events occur, (f) none occurs, (g) at most one occurs, (h) at most two occur.
An individual uses the following gambling system at Las Vegas. He bets $1 that the roulette wheel will come up red. If he wins, he quits. If he loses then he makes the same bet a second time only this time he bets $2; and then regardless of the outcome, quits. Assuming that he has a probability of
If P(E) 0.9 and P(F) 0.8, show that P(EF) 0.7. In general, show that P(EF) P(E) + P(F) - 1 This is known as Bonferroni's inequality.
We say that ECF if every point in E is also in F. Show that if ECF, then P(F) = P(E) + P(FE) P(E)
Show that P(UE) & PE) This is known as Boole's inequality. Hint: Either use Equation (1.2) and mathematical induction, or else show that UE, UF, where F, E, F, E, IE, and use property (iii) of a probability.
If two fair dice are tossed, what is the probability that the sum is i, i = 2, 3, ..., 12?
Let E and F be mutually exclusive events in the sample space of an experiment. Suppose that the experiment is repeated until either event E or event F occurs. What does the sample space of this new super experiment look like? Show that the probability that event E occurs before event F is
The dice game craps is played as follows. The player throws two dice, and if the sum is seven or eleven, then he wins. If the sum is two, three, or twelve, then he loses. If the sum is anything else, then he continues throwing until he either throws that number again (in which case he wins) or he
The probability of winning on a single toss of the dice is p. A starts, and if he fails, he passes the dice to B, who then attempts to win on her toss. They continue tossing the dice back and forth until one of them wins. What are their respective probabilities of winning?
Use Exercise 15 to show that P(E UF) = P(E) + P(F) - P(EF).
Suppose each of three persons tosses a coin. If the outcome of one of the tosses differs from the other outcomes, then the game ends. If not, then the persons start over and retoss their coins. Assuming fair coins, what is the probability that the game will end with the first round of tosses? If
Assume that each child that is born is equally likely to be a boy or a girl. If a family has two children, what is the probability that both are girls given that (a) the eldest is a girl, (b) at least one is a girl? *19. Two dice are rolled. What is the probability that at least one is a six? If
Three dice are thrown. What is the probability the same number appears on exactly two of the three dice?
Suppose that 5 percent of men and 0.25 percent of women are color- blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females.
A and B play until one has 2 more points than the other. Assuming that each point is independently won by A with probability p, what is the probability they will play a total of 2n points? What is the probability that A will win?
For events E, E2,..., E, show that P(EEE) = P(E)P(E, E,)P(E, E,E)... P(E,E,... E-1)
In an election, candidate A receives n votes and candidate B receives m votes, where n > m. Assume that in the count of the votes all possible orderings of the n+m votes are equally likely. Let P, denote the probability that from the first vote on A is always in the lead. Find (a) P2,1 (f) Pn, 2
Two cards are randomly selected from a deck of 52 playing cards. (a) What is the probability they constitute a pair (that is, that they are of the same denomination)? (b) What is the conditional probability they constitute a pair given that they are of different suits?
A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define events E, E, E, and E, as follows: E (the first pile has exactly 1 ace), E = [the second pile has exactly 1 ace], E, (the third pile has exactly 1 ace), E4 (the fourth pile has exactly 1
Suppose in Exercise 26 we had defined the events E,, i = 1, 2, 3, 4, by E =(one of the piles contains the ace of spades), E =(the ace of spaces and the ace of hearts are in different piles), E, (the ace of spades, the ace of hearts, and the E. ace of diamonds are in different piles), (all 4 aces
If the occurrence of B makes A more likely, does the occurrence of A make B more likely?
Suppose that P(E) = 0.6. What can you say about P(EIF) when (a) E and F are mutually exclusive? (b) EC F? (c) FCE?
Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability 0.7, whereas George, independently, hits the target with probability 0.4. (a) Given that exactly one shot hit the target, what is the probability that it was George's
What is the conditional probability that the first die is six given that the sum of the dice is seven?
Suppose all n men at a party throw their hats in the center of the room. Each man then randomly selects a hat. Show that the probability that none of the n men selects his own hat is 1 21 11 + 3! 4! ...(-1)" n! Note that as no this converges toe. Is this surprising?
In a class there are four freshman boys, six freshman girls, and six sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?
Mr. Jones has devised a gambling system for winning at roulette. When he bets, he bets on red, and places a bet only when the ten previous spins of the roulette have landed on a black number. He reasons that his chance of winning is quite large since the probability of eleven consecutive spins
A fair coin is continually flipped. What is the probability that the first four flips are (a) H, H, H, H? (b) T, H, H, H? (c) What is the probability that the pattern T, H, H, H occurs before the pattern H, H, H, H?
Consider two boxes, one containing one black and one white marble, the other, two black and one white marble. A box is selected at random and a marble is drawn at random from the selected box. What is the probability that the marble is black?
In Exercise 36, what is the probability that the first box was the one selected given that the marble is white?
Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in urn 2.A ball is then drawn from urn 2.It happens to be white. What is the probability that the transferred ball was white?
Stores A, B, and C have 50, 75, 100 employees, and respectively 50, 60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One employee resigns and this is a woman. What is the probability that she works in store C?
(a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. What is the probability that it is the fair coin? (b) Suppose that he flips the same coin a second time and again it shows heads. Now what is the prob-
In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling. (a) What is the probability that this rat is a pure black rat (as opposed to being a hybrid with one black and one brown gene)? (b) Suppose that when the black rat is mated
There are three coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin which comes up heads 75 percent of the time. When one of the three coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?
Suppose we have ten coins which are such that if the ith one is flipped then heads will appear with probability i/10, 1, 2,..., 10.When one of the coins is randomly selected and flipped, it shows heads. What is the conditional probability that it was the fifth coin?
Urn 1 has five white and seven black balls. Urn 2 has three white and twelve black balls. We flip a fair coin. If the outcome is heads, then a ball from urn 1 is selected, while if the outcome is tails, then a ball from urn 2 is selected. Suppose that a white ball is selected. What is the
An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn c additional balls of the same color are put in with it. Now suppose that we draw another ball. Show that the probability that the first ball drawn was black given that the second
Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this
Urn 1 has five white and seven black balls. Urn 2 has three white and twelve black balls. We flip a fair coin. If the outcome is heads, then a ball from urn 1 is selected, while if the outcome is tails, then a ball from urn 2 is selected. Suppose that a white ball is selected. What is the
Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this
An urn contains five red, three orange, and two blue balls. Two balls are randomly selected. What is the sample space of this experiment? Let X represent the number of orange balls selected. What are the possible values of X? Calculate P(X = 0].
In Exercise 2, if the coin is assumed fair, then, for n = 2, what arc the probabilities associated with the values that X can take on?
Suppose a die is rolled twice. What are the possible values that the following random variables can take on? (i) The maximum value to appear in the two rolls. (ii) The minimum value to appear in the two rolls. (iii) The sum of the two rolls. (iv) The value of the first roll minus the value of the
If the die in Exercise 4 is assumed fair, calculate the probabilities associated with the random variables in (i)-(iv).
Suppose five fair coins are tossed. Let E be the event that all coins land heads. Define the random variable I if E occurs IE = {0, if E occurs For what outcomes in the original sample space does I equal 1? What is PIE = 1}?
Suppose a coin having probability 0.7 of coming up heads is tossed three times. Let X denote the number of heads that appear in the three tosses. Determine the probability mass function of X.
Suppose the distribution function of X is given by (0, F(b) =, b
Suppose three fair dice are rolled. What is the probability that at most one six appears?
A ball is drawn from an urn containing three white and three black balls. After the ball is drawn, it is then replaced and another ball is drawn. This goes on indefinitely. What is the probability that of the first four balls drawn, exactly two are white?
On a multiple-choice exam with three possible answers for each of the five questions, what is the probability that a student would get four or more correct answers just by guessing?
An individual claims to have extrasensory perception (ESP). As a test, a fair coin is flipped ten times, and he is asked to predict in advance the outcome. Our individual gets seven out of ten correct. What is the prob- ability he would have done at least this well if he had no ESP? (Explain why
Suppose X has a binomial distribution with parameters 6 and . Show that X3 is the most likely outcome.
Let X be binomially distributed with parameters n and p. Show that as k goes from 0 to n, P(X = k) increases monotonically, then decreases monotonically reaching its largest value. (a) in the case that (n + 1)p is an integer, when k equals either (n+1)p 1 or (n + 1)p, (b) in the case that (n + 1)p
An airline knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can only hold 50 passengers. What is the probability that there will be a seat available for every passenger that shows up?
Suppose that an experiment can result in one of r possible outcomes, the ith outcome having probability pi, i = 1,...,, E P = 1.If n of these experiments are performed, and if the outcome of any one of the n does not affect the outcome of the other n 1 experiments, then show that the probability
Show that when r2 the multinomial reduces to the binomial.
In Exercise 17, let X, denote the number of times the ith outcome appears, 1,..,. What is the probability mass function of
A television store owner figures that 50 percent of the customers entering his store will purchase an ordinary television set, 20 percent will purchase a color television set, and 30 percent will just be browsing. If five customers enter his store on a certain day, what is the probability that two
In Exercise 20, what is the probability that our store owner sells three or more televisions on that day?
If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.
A coin having a probability p of coming up heads is successively flipped until the rth head appears. Argue that X, the number of flips required, will be n, nzr, with probability P(X = n] = = ( - )p(1 - p)*** r-1 This is known as the negative binomial distribution. nr Hint: How many successes must
The probability mass function of X is given by (r+k p(k) = (+-)p(1 - p)* k = 0, 1, ... Give a possible intepretation of the random variable X. Hint: See Exercise 23.In Exercises 25 and 26, suppose that two teams are playing a series of games, each of which is independently won by team A with
If i 4, find the probability that a total of 7 games are played. Also show that this probability is maximized when p = 1/2.
Find the expected number of games that are played when (a) i = 2.(b) i = 3.In both cases, show that this number is maximized when p = 1/2.
A fair coin is independently flipped n times, k times by A and n - k times by B. Show that the probability that A and B flip the same number of heads is equal to the probability that there are a total of k heads.
Suppose that we want to generate a random variable X that is equally likely to be either 0 or 1, and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some (unknown) probability p. Consider the following procedure: 1.Flip the coin, and let 0,, either heads or
Consider n independent flips of a coin having probability p of landing heads. Say a changeover occurs whenever an outcome differs from the one preceding it. For instance, if the results of the flips are H HTHTHHT, then there are a total of 5 changeovers. If p = 1/2, what is the probability there
Let X be a Poisson random variable with parameter A. Show that PIX ) increases monotonically and then decreases monotonically as i increases, reaching its maximum when i is the largest integer not exceeding 2.Hint: Consider P(X = i\/P(X = i - 1].
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