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theory of probability

Questions and Answers of Theory Of Probability

22. 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
21. 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
20. Three dice are thrown. What is the probability the same number appears on exactly two of the three dice?
19. Two dice are rolled. What is the probability that at least one is a six? If the two faces are different, what is the probability that at least one is a six?
18. Assume that each child who 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
17. 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.
16. Use Exercise 15 to show that P(E ∪ F) = P(E) + P(F) − P(EF).
15. Argue that E = EF ∪ EFc, E ∪ F = E ∪ FEc.
14. 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
13. The dice game craps is played as follows. The player throws two dice, and if the sum is seven or eleven, then she wins. If the sum is two, three, or twelve, then she loses. If the sum is anything
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
11. If two fair dice are tossed, what is the probability that the sum is i, i =2, 3,..., 12?
10. Show that Pn i=1 Ein i=1 P(Ei)This is known as Boole’s inequality.Hint:Either use Equation (1.2) and mathematical induction, or else show that n i=1 Ei =n i=1 Fi, where F1 = E1, Fi = Ei
9. We say that E ⊂ F if every point in E is also in F. Show that if E ⊂ F, then P(F) = P(E) + P(FEc) P(E)
8. 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.
7. Show that (E ∪ F)c = EcFc.
6. Show that E(F ∪ G) = EF ∪ EG.
5. 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 ?
4. 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 occur,(c) at least one event occurs,(d) at least two events occur,(e) all three
3. 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, what is the probability that it will be tossed exactly four times?
2. Repeat Exercise 1 when the second marble is drawn without replacing the first marble.
1. 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
Let Y denote an exponential random variable with rate λ that is independent of the continuous-time Markov chain {X(t)} and let P¯i j = P{X(Y ) = j|X(0) = i}(a) Show that P¯i j = 1 vi + λk qik
Consider the two-state continuous-time Markov chain. Starting in state 0, find Cov[X(s), X(t)].
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].
In Example 6.20, we computed m(t) = E[O(t)], the expected occupation time in state 0 by time t for the two-state continuous-time Markov chain starting in state 0. Another way of obtaining this
For the continuous-time Markov chain of Exercise 3 present a uniformized version.
Consider a sequential queueing model with three servers, where customers arrive at server 1 in accordance with a Poisson process with rate λ. After completion at server 1 the customer then moves to
In Example 6.22 explain why we would have known before analyzing Example 6.22 that the limiting probability there are j customers with serveri is(λ/μi)j(1−λ/μi), i = 1, 2, j 0. (What we would
Show in Example 6.22 that the limiting probabilities satisfy Equations (6.33),(6.34), and (6.35).
Consider a continuous-time Markov chain with states 1,..., n, which spends an exponential time with rate vi in state i during each visit to that state and is then equally likely to go to any of the
Suppose in Exercise 38 that an entering customer is served by the server who has been idle the shortest amount of time.(a) Define states so as to analyze this model as a continuous-time Markov
Consider an n server system where the service times of server i are exponentially distributed with rate μi, i = 1,..., n. Suppose customers arrive in accordance with a Poisson process with rate λ,
A hospital accepts k different types of patients, where type i patients arrive according to a Poisson proccess with rate λi , with these k Poisson processes being independent. Type i patients spend
Consider a system of n components such that the working times of component i, i = 1,..., n, are exponentially distributed with rate λi . When a component fails, however, the repair rate of component
Consider a time reversible continuous-time Markov chain having infinitesimal transition rates qi j and limiting probabilities {Pi}. Let A denote a set of states for this chain, and consider a new
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
Consider two M/M/1 queues with respective parameters λi, μi,i = 1, 2. Suppose they share a common waiting room that can hold at most three customers.That is, whenever an arrival finds her server
Customers arrive at a two-server station in accordance with a Poisson process having rate λ. Upon arriving, they join a single queue. Whenever a server completes a service, the person first in line
A total of N customers move about among r servers in the following manner.When a customer is served by server i, he then goes over to server j, j = i, with probability 1/(r − 1). If the server he
Consider a graph with nodes 1, 2,..., n and the n 2arcs (i, j),i = j,i, j, =1,..., n. (See Section 3.6.2 for appropriate definitions.) Suppose that a particle moves along this graph as follows:
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
If {X(t)} and {Y (t)} are independent continuous-time Markov chains, both of which are time reversible, show that the process {X(t), Y (t)} is also a time reversible Markov chain.
In the M/M/s queue if you allow the service rate to depend on the number in the system (but in such a way so that it is ergodic), what can you say about the output process? What can you say when the
Consider an ergodic M/M/s queue in steady state (that is, after a long time) and argue that the number presently in the system is independent of the sequence of past departure times. That is, for
Customers arrive at a service station, manned by a single server who serves at an exponential rate μ1, at a Poisson rate λ. After completion of service the custome?
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
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
Customers arrive at a single-server queue in accordance with a Poisson process having rate λ. However, an arrival that finds n customers already in the system will only join the system with
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 μ. Now find the
There are two machines, one of which is used as a spare. A working machine will function for an exponential time with rate λ and will then fail. Upon failure, it is immediately replaced by the other
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 λi,i = 1, 2. When machine i fails, it requires an
After being repaired, a machine functions for an exponential time with rate λand then fails. Upon failure, a repair process begins. The repair process proceeds sequentially through k distinct
Each time a machine is repaired it remains up for an exponentially distributed time with rate λ. It then fails, and its failure is either of two types. If it is a type 1 failure, then the time to
The following problem arises in molecular biology. The surface of a bacterium consists of several sites at which foreign molecules—some acceptable and some not—become attached. We consider a
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
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
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
Each individual in a biological population is assumed to give birth at an exponential rate λ, and to die at an exponential rate μ. In addition, there is an exponential rate of increase θ due to
Consider a Yule process starting with a single individual—that is, suppose X(0) = 1. Let Ti denote the time it takes the process to go from a population of size i to one of size i + 1.(a) Argue
Consider two machines. Machine i operates for an exponential time with rate λi and then fails; its repair time is exponential with rate μi,i = 1, 2. The machines act independently of each other.
The birth and death process with parameters λn = 0 and μn = μ, n > 0 is called a pure death process. Find Pi j(t).
Consider two machines, both of which have an exponential lifetime with mean 1/λ. There is a single repairman that can service machines at an exponential rateμ. Set up the Kolmogorov backward
Individuals join a club in accordance with a Poisson process with rate λ. Each new member must pass through k consecutive stages to become a full member of the club. The time it takes to pass
Consider a birth and death process with birth rates λi = (i + 1)λ,i 0, and death rates μi = iμ,i 0.(a) Determine the expected time to go from state 0 to state 4.(b) Determine the expected time to
There are N individuals in a population, some of whom have a certain infection that spreads as follows. Contacts between two members of this population occur in accordance with a Poisson process
Potential customers arrive at a single-server station in accordance with a Poisson process with rate λ. However, if the arrival finds n customers already in the station, then he will enter the
Consider two machines that are maintained by a single repairman. Machine i functions for an exponential time with rate μi before breaking down, i = 1, 2.The repair times (for either machine) are
Suppose that a one-celled organism can be in one of two states—either A or B. An individual in state A will change to state B at an exponential rate α; an individual in state B divides into two
A population of organisms consists of both male and female members. In a small colony any particular male is likely to mate with any particular female in any time interval of length h, with
22. Let X be a Poisson random variable with mean 20. (a) Use the Markov inequality to obtain an upper bound on p = P(X26] (b) Use the one-sided Chebyshev inequality to obtain an upper bound on p. (e)
21. Would the results of Example 5f change if the investor were allowed to divide her money and invest the fractiona, 0 < a
20. Let X be a nonnegative random variable. Prove that E[X] (E[X])=(E[X])=...
19. If X is a nonnegative random variable with mean 25, what can be said about:(a) E[X³];(b) E[√X];(c) E[log X];(d) E[e-X]?
18. A lake contains 4 distinct types of fish. Suppose that each fish caught is equally likely to be any one of these types. Let Y denote the number of fish that need be caught to obtain at least one
17. Repeat part (a) of Problem 2 when it is known that the variance of a student's test score is equal to 25.
16. Redo Example 5b under the assumption that the number of man-woman pairs is (approximately) normally distributed. Does this seem like a reasonable supposition?
15. An insurance company has 10,000 automobile policyholders. The expected yearly claim per policyholder is $240 with a standard deviation of $800.Approximate the probability that the total yearly
14. A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its
13. Student scores on exams given by a certain instructor have mean 74 and standard deviation 14. This instructor is about to give two exams, one to a class of size 25 and the other to a class of
12. We have 100 components that we will put in use in a sequential fashion.That is, component 1 is initially put in use, and upon failure it is replaced by component 2, which is itself replaced upon
11. Many people believe that the daily change of price of a company's stock on the stock market is a random variable with mean 0 and variance σ². That is, if Yt represents the price of the stock on
10. Civil engineers believe that W, the amount of weight (in units of 1000 pounds)that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with mean
9. If X is a gamma random variable with parameters (n, 1) how large need n be so that$$P[|X - n| > .01] < .01?$$
8. In Problem 7 suppose that it takes a random time, uniformly distributed over(0, 5), to replace a failed bulb. What is the probability that all bulbs have failed by time 550?
7. One has 100 light bulbs whose lifetimes are independent exponentials with mean 5 hours. If the bulbs are used one at a time, with a failed bulb being replaced immediately by a new one, what is the
6. A die is continually rolled until the total sum of all rolls exceeds 300. What is the probability that at least 80 rolls are necessary?
5. Fifty numbers are rounded off to the nearest integer and then summed. If the individual round-off errors are uniformly distributed over (-5, 5) what is the probability that the resultant sum
4. Let X1, X20 be independent Poisson random variables with mean 1.(a) Use the Markov inequality to obtain a bound on$$P[\sum_{i=1}^{20} X_i > 15]$$(b) Use the central limit theorem to
3. Use the central limit theorem to solve part (c) of Problem 2.
2. From past experience a professor knows that the test score of a student taking her final examination is a random variable with mean 75. (a) Give an upper bound for the probability that a student's
1. Suppose that X is a random variable with mean and variance both equal to 20. What can be said about P[0 X= 40)?
14. In Self-Test Problem 1 we showed how to use the value of a uniform (0, 1) random variable (commonly called a random number) to obtain the value of a random variable whose mean is equal to the
13. Each coin in a bin has a value attached to it. Each time that a coin with value p is flipped it lands on heads with probability p. When a coin is randomly chosen from the bin, its value is
12. A deck of 52 cards is shuffled and a bridge hand of 13 cards is dealt out. Let X and Y denote, respectively, the number of aces and the number of spades in the dealt hand. (a) Show that X and Y
11. The nine players on a basketball team consist of 2 centers, 3 forwards, and 4 backcourt players. If the players are paired up at random into three groups of size 3 each, find the (a) expected
10. Individuals 1 through n, n > 1, are to be recruited into a firm in the following manner. Individual 1 starts the firm and recruits individual 2. Individuals 1 and 2 will then compete to recruit
9. Suppose in Self-Test Problem 3 that the 20 people are to be seated at seven tables, three of which have 4 seats and four of which have 2 seats. If the people are randomly seated, find the expected
8. Let X be a Poisson random variable with mean A. Show that if A is not too small, then$$Var(\sqrt{X}) \approx 25$$HINT: Use the result of Theoretical Exercise 4 to approximate E[VX].
7. Suppose that k of the balls numbered 1, 2,..., n, where nk, are randomly chosen. Let X denote the maximum numbered ball chosen. Also, let R denote the number of the n k unchosen balls that have

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