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probability and stochastic modeling

An Introduction To Stochastic Modeling 4th Edition Mark A. Pinsky, Samuel Karlin - Solutions

3.4.4 A coin is tossed repeatedly until two successive heads appear. Find the mean number of tosses required.Hint: Let Xn be the cumulative number of successive heads. The state space is 0;1; 2, and the transition probability matrix isDetermine the mean time to reach state 2 starting from state 0
3.4.3 Consider the Markov chain whose transition probability matrix is given by(a) Starting in state 1, determine the probability that the Markov chain ends in state 0.(b) Determine the mean time to absorption. 0 1 2 3 0 100 0 1 0.1 0.6 0.1 0.2 P = 2 0.2 0.3 0.4 0.1 3 000 1
3.4.2 Consider the Markov chain whose transition probablity matrix is given by(a) Starting in state 1, determine the probability that the Markov chain ends in state 0.(b) Determine the mean time to absorption. 012 0 10 0 P 1 0.1 0.6 0.3 20 00 1
3.4.1 Find the mean time to reach state 3 starting from state 0 for the Markov chain whose transition probability matrix is 0 1 2 3 P = 0 0.4 0.3 0.2 0.1| 10 0.7 0.2 0.1 23 3 0 0 0.9 0.1 000 1
3.3.10 Consider a discrete-time, periodic review inventory model and let n be the total demand in period n, and let Xn be the inventory quantity on hand at the end-of-period n. An .s;S/ inventory policy is used: If the end-of-period stock is not greater than s, then a quantity is instantly
3.3.9 Suppose that two urns A and B contain a total of N balls. Assume that at time t, there are exactly k balls in A. At time tC1, a ball and an urn are chosen with probability depending on the contents of the urn (i.e., a ball is chosen from A with probability k=N or from B with probability .N
3.3.8 Two urns A and B contain a total of N balls. Assume that at time t, there were exactly k balls in A. At time tC1, an urn is selected at random in proportion to its contents (i.e., A is chosen with probability k=N and B is chosen with probability .N ????k/=N). Then, a ball is selected from A
3.3.7 A component in a system is placed into service, where it operates until its failure, whereupon it is replaced at the end of the period with a new component having statistically identical properties, and the process repeats. The probability that a component lasts for k periods is k, for k D
3.3.6 Two teams, A and B, are to play a best of seven series of games. Suppose that the outcomes of successive games are independent, and each is won by A with probability p and won by B with probability 1????p. Let the state of the system be represented by the pair .a;b/, where a is the number of
3.3.5 You are going to successively flip a quarter until the pattern HHT appears, that is, until you observe two successive heads followed by a tails. In order to calculate some properties of this game, you set up a Markov chain with the following states: 0;H;HH, and HHT, where 0 represents the
3.3.4 Consider the queueing model of Section 3.4. Now, suppose that at most a single customer arrives during a single period, but that the service time of a customer is a random variable Z with the geometric probability distribution PrfZ D kg D .1????/k????1 for k D 1;2; : : : :Specify the
3.3.3 Consider the inventory model of Section 3.3.1. Suppose that unfulfilled demand is not back ordered but is lost.(a) Set up the corresponding transition probability matrix for the end-ofperiod inventory level Xn.(b) Express the long run fraction of lost demand in terms of the demand
3.3.2 Three fair coins are tossed, and we let X1 denote the number of heads that appear. Those coins that were heads on the first trial (there were X1 of them)we pick up and toss again, and now we let X2 be the total number of tails, including those left from the first toss. We toss again all coins
3.3.1 An urn contains six tags, of which three are red and three are green. Two tags are selected from the urn. If one tag is red and the other is green, then the selected tags are discarded and two blue tags are returned to the urn. Otherwise, the selected tags are resumed to the urn. This process
3.3.5 An urn initially contains a single red ball and a single green ball. A ball is drawn at random, removed, and replaced by a ball of the opposite color, and this process repeats so that there are always exactly two balls in the urn. Let Xn be the number of red balls in the urn after n draws,
3.3.4 Consider the inventory model of Section 3.3.1. Suppose that S D 3 and that the probability distribution for demand is Prf D 0g D 0:1; Prf D 1g D 0:4;Prf D 2g D 0:3, and Prf D 3g D 0:2. Set up the corresponding transition probability matrix for the end-of-period inventory level Xn.
3.3.3 Consider the inventory model of Section 3.3.1. Suppose that S D 3. Set up the corresponding transition probability matrix for the end-of-period inventory level Xn.
3.3.2 Consider two urns A and B containing a total of N balls. An experiment is performed in which a ball is selected at random (all selections equally likely) at time t.t D 1;2; : : :/ from among the totality of N balls. Then, an urn is selected at random (A is chosen with probability p and B is
3.3.1 Consider a spare parts inventory model in which either 0; 1, or 2 repair parts are demanded in any period, withand suppose s D 0 and S D 3. Determine the transition probability matrix for the Markov chain fXng, where Xn is defined to be the quantity on hand at the end-of-period n.
3.2.5 A Markov chain has the transition probability matrixThe Markov chain starts at time zero in state X0 D 0. Let T D minfn 0IXn D 2g be the first time that the process reaches state 2. Eventually, the process will reach and be absorbed into state 2. If in some experiment we observed such a
3.2.4 Suppose Xn is a two-state Markov chain whose transition probability matrix isThen, Zn D .Xn????1;Xn/ is a Markov chain having the four states .0;0/; .0;1/;.1;0/, and .1;1/. Determine the transition probability matrix. 0 1-a P = 1-B
3.2.3 Let Xn denote the quality of the nth item produced by a production system with Xn D 0 meaning “good” and Xn D 1 meaning “defective.” Suppose that Xn evolves as a Markov chain whose transition probability matrix isWhat is the probability that the fourth item is defective given that the
3.2.2 Consider the problem of sending a binary message, 0 or 1, through a signal channel consisting of several stages, where transmission through each stage is subject to a fixed probability of error . Let X0 be the signal that is sent, and let Xn be the signal that is received at the nth stage.
3.2.1 Consider the Markov chain whose transition probability matrix is given bySuppose that the initial distribution is pi D 14 for i D 0;1;2; 3. Show that PrfXn D kg D 1 4 ; k D 0;1;2; 3, for all n. Can you deduce a general result from this example? 0 1 2 3 00.4 0.3 0.2 0.1 1 0.1 0.4 0.3 0.2 P = 2
3.2.6 A Markov chain X0;X1;X2; : : : has the transition probability matrixand initial distribution p0 D 0:5 and p1 D 0:5. Determine the probabilities PrfX2 D 0g and PrfX3 D 0g. 012 00.3 0.2 0.5 P 1 0.5 0.1 0.4 2 0.5 0.2 0.3
3.2.5 A Markov chain X0;X1;X2; : : : has the transition probability matrixDetermine the conditional probabilities PrfX3 D 1jX1 D 0g and PrfX2 D 1jX0 D 0g: 012 0 0.1 0.1 0.8 P=1 P 1 0.2 0.2 0.6 2 0.3 0.3 0.4
3.2.4 A Markov chain X0;X1;X2; : : : has the transition probability matrixIf it is known that the process starts in state X0 D 1, determine the probability PrfX2 D 2g. 012 0 0.6 0.3 0.1 P 1 0.3 0.3 0.4 2 0.4 0.1 0.5
3.2.3 A Markov chain X0;X1;X2; : : : has the transition probability matrixDetermine the conditional probabilities PrfX3 D 1jX0 D 0g and PrfX4 D 1jX0 D 0g: 0 00.7 0 0.7 P 1 0 12 0.2 0.1 P=1 0.6 0.4 2 0.5 0 0.5
3.2.2 A particle moves among the states 0;1;2 according to a Markov process whose transition probability matrix isLet Xn denote the position of the particle at the nth move. Calculate PrfXn D 0jX0 D 0g for n D 0;1;2;3; 4. 012 21212 -12 0 12 1212 P = 1 2
3.2.1 A Markov chain fXng on the states 0;1;2 has the transition probability matrix(a) Compute the two-step transition matrix P2.(b) What is PrfX3 D 1jX1 D 0g?(c) What is PrfX3 D 1jX0 D 0g? 012 00.1 0.2 0.2 0.7 P 1 0.2 0.2 0.6 = 2 0.6 0.1 0.3
3.1.4 The random variables 1; 2; : : : are independent and with the common probability mass functionSet X0 D 0, and let Xn D maxf1; : : : ; ng be the largest observed to date. Determine the transition probability matrix for the Markov chain fXng. k = 0 1 2 3 Pr{k} = 0.1 0.3 0.2 0.4
3.1.3 Consider a sequence of items from a production process, with each item being graded as good or defective. Suppose that a good item is followed by another good item with probability and is followed by a defective item with probability 1????. Similarly, a defective item is followed by another
3.1.2 Consider the problem of sending a binary message, 0 or 1, through a signal channel consisting of several stages, where transmission through each stage is subject to a fixed probability of error . Suppose that X0 D 0 is the signal that is sent and let Xn be the signal that is received at the
3.1.1 A simplified model for the spread of a disease goes this way: The total population size is N D 5, of which some are diseased and the remainder are healthy.During any single period of time, two people are selected at random from the population and assumed to interact. The selection is such
3.1.5 A Markov chain X0;X1;X2; : : : has the transition probability matrixand initial distribution p0 D 0:5 and p1 D 0:5. Determine the probabilities PrfX0 D 1;X1 D 1;X2 D 0g and PrfX1 D 1;X2 D 1;X3 D 0g: 012 0 0.3 0.2 0.5 P 1 0.5 0.1 0.4 2 0.5 0.2 0.3
3.1.4 A Markov chain X0;X1;X2; : : : has the transition probability matrixDetermine the conditional probabilities PrfX1 D 1;X2 D 1jX0 D 0g and PrfX2 D 1;X3 D 1jX1 D 0g: 012 0 0.1 0.1 0.8 P 1 0.2 0.2 0.6 2 0.3 0.3 0.4 20.3
3.1.3 A Markov chain X0;X1;X2; : : : has the transition probability matrixIf it is known that the process starts in state X0 D 1, determine the probability PrfX0 D 1;X1 D 0;X2 D 2g. 012 0 0.6 0.3 0.1| P 10.3 0.3 0.4 2 0.4 0.1 0.5
3.1.2 A Markov chain X0;X1;X2; : : : has the transition probability matrixDetermine the conditional probabilities PrfX2 D 1;X3 D 1jX1 D 0g and PrfX1 D 1;X2 D 1jX0 D 0g: 012 0 0.7 0.2 0.1 P=1 0 0.6 0.4 2 0.5 0 0.5
3.1.1 A Markov chain X0;X1; : : : on states 0, 1, 2 has the transition probability matrixand initial distribution p0 D PrfX0 D 0g D 0:3;p1 D PrfX0 D 1g D 0:4, and p2 D PrfX0 D 2g D 0:3. Determine PrfX0 D 0;X1 D 1;X2 D 2g. 0 1 2 0 0.1 0.2 0.7 P=1 P 10.9 0.1 0 2 0.1 0.8 0.1
2.5.5 Consider a stochastic process that evolves according to the following laws: If Xn D 0, then XnC1 D 0, whereas if Xn > 0, then(a) Show that Xn is a nonnegative martingale.(b) Suppose that X0 D i > 0. Use the maximal inequality to bound PrfXn N for some n 0jX0 D ig:Note: Xn represents
2.5.4 Let 1; 2; : : : be independent Bernoulli random variables with parameter p;0
2.5.3 Let S0 D 0, and for n 1, let Sn D "1 C C"n be the sum of n independent random variables, each exponentially distributed with mean E["] D 1. Show that Xn D 2n exp.????Sn/; n 0 defines a martingale.
2.5.2 Let U1;U2; : : : be independent random variables each uniformly distributed over the interval .0; 1]. Show that X0 D 1 and Xn D 2nU1 Un for n D 1;2; : : :defines a martingale.
2.5.1 Use the law of total probability for conditional expectations E[EfXjY;ZgjZ] D E[XjZ] to show E[XnC2jX0; : : : ;Xn] D E[EfXnC2jX0; : : : ;XnC1gjX0; : : : ;Xn]:Conclude that when Xn is a martingale, E[XnC2jX0; : : : ;Xn] D Xn:
2.5.3 Let be a random variable with mean and standard deviation . Let X D. ????/2. Apply Markov’s inequality to X to deduce Chebyshev’s inequality: Pr{ ) for any > 0.
2.5.2 Let X be a Bernoulli random variable with parameter p. Compare PrfX 1g with the Markov inequality bound.
2.5.1 Let X be an exponentially distributed random variable with mean E[X] D 1. For x D 0:5; 1, and 2, compare PrfX > xg with the Markov inequality bound E[X]=x.
2.4.8 Let X and Y have the normal density given in Chapter 1, in (1.47). Show that the conditional density function for X, given that Y D y, is normal with moments xx =x+ and OXY = 0x1-p (Art).
2.4.7 Suppose that X and Y are independent random variables, each having the same exponential distribution with parameter . What is the conditional probability density function for X, given that Z D X CY D z?
2.4.6 Let X0;X1;X2; : : : be independent identically distributed nonnegative random variables having a continuous distribution. Let N be the first index k for which Xk > X0. That is, N D 1 if X1 > X0;N D 2 if X1 X0 and X2 > X0, etc. Determine the probability mass function for N and the mean E[N].
2.4.5 Let X and Y be jointly distributed random variables whose joint probability mass function is given in the following table:Show that the covariance between X and Y is zero even though X and Y are not independent. -1 -1 x 0 1 27 -19 - 0 2 y 0 1 67 -19 0 20 0 p(x, y) Pr{X=x, Y=y}
2.4.4 Suppose X and Y are independent random variables having the same Poisson distribution with parameter , but where is also random, being exponentially distributed with parameter . What is the conditional distribution for X given that X CY D n?
2.4.3 Let X have a Poisson distribution with parameter > 0. Suppose itself is random, following an exponential density with parameter .(a) What is the marginal distribution of X?(b) Determine the conditional density for given X D k.
2.4.2 Let N have a Poisson distribution with parameter > 0. Suppose that, conditioned on N D n, the random variable X is binomially distributed with parameters N D n and p. Set Y D N ????X. Show that X and Y have Poisson distributions with respective parameters p and .1????p/ and that X and Y
2.4.1 Suppose that the outcome X of a certain chance mechanism depends on a parameter p according to PrfX D 1g D p and PrfX D 0g D 1????p, where 0 p 1. Suppose that p is chosen at random, uniformly distributed over the unit interval[0; 1], and then, that two independent outcomes X1 and X2 are
2.4.5 Let U be uniformly distributed over the interval [0;L] where L follows the gamma density fL.x/ D xe????x for x 0. What is the joint density function of U and V D L????U?
2.4.4 Suppose X and Y are independent random variables, each exponentially distributed with parameter . Determine the probability density function for ZDX=Y.
2.4.3 A random variable T is selected that is uniformly distributed over the interval.0; 1]. Then, a second random variable U is chosen, uniformly distributed on the interval .0;T]. What is the probability that U exceeds 1 2 ?
2.4.2 Suppose that three components in a certain system each function with probability p and fail with probability 1????p, each component operating or failing independently of the others. But the system is in a random environment so that p is itself a random variable. Suppose that p is uniformly
2.4.1 Suppose that three contestants on a quiz show are each given the same question and that each answers it correctly, independently of the others, with probability p. But the difficulty of the question is itself a random variable, so let us suppose, for the sake of illustration, that p is
2.3.5 To form a slightly different random sum, let 0; 1; : : : be independent identically distributed random variables and let N be a nonnegative integer-valued random variable, independent of 0; 1; : : : : The first two moments areDetermine the mean and variance of the random sum E[k] = E[N] v,
2.3.4 Suppose 1; 2; : : : are independent and identically distributed random variables having mean and variance 2. Form the random sum SN D 1 C CN.(a) Derive the mean and variance of SN when N has a Poisson distribution with parameter .(b) Determine the mean and variance of SN when N
2.3.3 Suppose that 1; 2; : : : are independent and identically distributed with Prfk D1g D 1 2 . Let N be independent of 1; 2; : : : and follow the geometric probability mass functionwhere 0 1 C CN.(a) Determine the mean and variance of Z.(b) Evaluate the higher moments m3 D E Z3and m4 D E
2.3.2 For each given p, let Z have a binomial distribution with parameters p and N.Suppose that N is itself binomially distributed with parameters q and M. Formulate Z as a random sum and show that Z has a binomial distribution with parameters pq and M.
2.3.1 The following experiment is performed: An observation is made of a Poisson random variable N with parameter . Then N independent Bernoulli trials are performed, each with probability p of success. Let Z be the total number of successes observed in the N trials.(a) Formulate Z as a random sum
2.3.5 The number of accidents occurring in a factory in a week is a Poisson random variable with mean 2. The number of individuals injured in different accidents is independently distributed, each with mean 3 and variance 4. Determine the mean and variance of the number of individuals injured in a
2.3.4 A six-sided die is rolled, and the number N on the uppermost face is recorded.From a jar containing 10 tags numbered 1;2; : : : ;10 we then select N tags at random without replacement. Let X be the smallest number on the drawn tags.Determine PrfX D 2g and E[X].
2.3.3 Suppose that upon striking a plate a single electron is transformed into a number N of electrons, where N is a random variable with mean and standard deviation. Suppose that each of these electrons strikes a second plate and releases further electrons, independently of each other and each
2.3.2 Six nickels are tossed, and the total number N of heads is observed. Then N dimes are tossed, and the total number Z of tails among the dimes is observed.Determine the mean and variance of Z. What is the probability that Z D 2?
2.3.1 A six-sided die is rolled, and the number N on the uppermost face is recorded.Then a fair coin is tossed N times, and the total number Z of heads to appear is observed. Determine the mean and variance of Z by viewing Z as a random sum of N Bernoulli random variables. Determine the probability
2.2.2 Consider a pair of dice that are unbalanced by the addition of weights in the following manner: Die #1 has a small piece of lead placed near the four side, causing the appearance of the outcome 3 more often than usual, while die #2 is weighted near the three side, causing the outcome 4 to
2.2.1 Let X1;X2; : : : be independent identically distributed positive random variables whose common distribution function is F. We interpret X1;X2; : : : as successive bids on an asset offered for sale. Suppose that the policy is followed of accepting the first bid that exceeds some prescribed
2.2.3 Determine the win probability when the dice are shaved on the 1–6 faces and pC D 0:206666 and p???? D 0:146666 .
2.2.2 Verify the win probability of 0.5029237 by substituting from (2.21) into (2.20).
2.2.1 A red die is rolled a single time. A green die is rolled repeatedly. The game stops the first time that the sum of the two dice is either 4 or 7. What is the probability that the game stops with a sum of 4?
2.1.10 Do men have more sisters than women have? In a certain society, all married couples use the following strategy to determine the number of children that they will have: If the first child is a girl, they have no more children. If the first child is a boy, they have a second child. If the
2.1.9 Let N have a Poisson distribution with parameter D 1. Conditioned on N D n, let X have a uniform distribution over the integers 0;1; : : : ;nC1. What is the marginal distribution for X?
2.1.8 Initially an urn contains one red and one green ball. A ball is drawn at random from the urn, observed, and then replaced. If this ball is red, then an additional red ball is placed in the urn. If the ball is green, then a green ball is added. A second ball is drawn. Find the conditional
2.1.7 The probability that an airplane accident that is due to structural failure is correctly diagnosed is 0.85, and the probability that an airplane accident that is not due to structural failure is incorrectly diagnosed as being due to structural failure is 0.35. If 30% of all airplane accidents
2.1.6 A dime is tossed repeatedly until a head appears. Let N be the trial number on which this first head occurs. Then, a nickel is tossed N times. Let X count the number of times that the nickel comes up tails. Determine PrfXD0g;PrfXD1g, and E[X].
2.1.5 A nickel is tossed 20 times in succession. Every time that the nickel comes up heads, a dime is tossed. Let X count the number of heads appearing on tosses of the dime. Determine PrfX D 0g.
2.1.4 Suppose that X has a binomial distribution with parameters p D 1 2 and N, where N is also random and follows a binomial distribution with parameters q D 14 and M D 20. What is the mean of X?
2.1.3 Let X and Y denote the respective outcomes when two fair dice are thrown. Let U D minfX;Yg;V D maxfX;Yg, and S D U CV;T D V ????U.(a) Determine the conditional probability mass function for U given V D v.(b) Determine the joint mass function for S and T.
2.1.2 A card is picked at random from N cards labeled 1;2; : : : ;N, and the number that appears is X. A second card is picked at random from cards numbered 1;2; : : : ;X and its number is Y. Determine the conditional distribution of X given Y D y, for y D 1;2; : : : :
2.1.1 Let M have a binomial distribution with parameters N and p. Conditioned on M, the random variable X has a binomial distribution with parametersM and .(a) Determine the marginal distribution for X.(b) Determine the covariance between X and Y D M????X.
2.1.6 Suppose U and V are independent and follow the geometric distributionDefine the random variable Z D U CV.(a) Determine the joint probability mass function pU;Z.u; z/D PrfUDu;ZDzg.(b) Determine the conditional probability mass function for U given that Z D n. p(k) = p(1-p) for k=0,1,....
2.1.5 Let X be a Poisson random variable with parameter . Find the conditional mean of X given that X is odd.
2.1.4 A six-sided die is rolled, and the number N on the uppermost face is recorded.From a jar containing 10 tags numbered 1;2; : : : ; 10, we then select N tags at random without replacement. Let X be the smallest number on the drawn tags.Determine PrfX D 2g.
2.1.3 A poker hand of five cards is dealt from a normal deck of 52 cards. Let X be the number of aces in the hand. Determine PrfX > 1jX 1g. This is the probability that the hand contains more than one ace, given that it has at least one ace.Compare this with the probability that the hand contains
2.1.2 Four nickels and six dimes are tossed, and the total number N of heads is observed. If N D 4, what is the conditional probability that exactly two of the nickels were heads?
2.1.1 I roll a six-sided die and observe the number N on the uppermost face. I then toss a fair coin N times and observe X, the total number of heads to appear. What is the probability that N D 3 and X D 2? What is the probability that X D 5? What is E[X], the expected number of heads to appear?
1.5.9 A flashlight requires two good batteries in order to shine. Suppose, for the sake of this academic exercise, that the lifetimes of batteries in use are independent random variables that are exponentially distributed with parameter D 1.Reserve batteries do not deteriorate. You begin with
1.5.8 Let U1;U2; : : : ;Un be independent uniformly distributed random variables on the unit interval [0; 1]. Define the minimum Vn D minfU1;U2; : : : ;Ung.(a) Show that PrfVn > vg D .1????v for 0 v 1.(b) Let Wn D nVn. Show that PrfWn > wg D [1????.w=n/]n for 0 w n, and thus lim
1.5.7 Let X1;X2; : : : , Xn be independent random variables that are exponentially distributed with respective parameters 1;2; : : : , n. Identify the distribution of the minimum V D minfX1;X2; : : : ;Xng.Hint: For any real number v, the event fV > vg is equivalent to fX1 > v;X2 >v; : : : ;Xn >
1.5.6 Determine the upper tail probabilities PrfV > tg and mean E[V] for a random variable V having the exponential densitywhere is a fixed positive parameter. Jo fv (v) = for v < 0, The for v 0,
1.5.5 Show thatfor a nonnegative random variable W. E[W]= 2y[1-Fw(y)]dy 0
1.5.4 Let V be a continuous random variable taking both positive and negative values and whose mean exists. Derive the formula 0 E[V] = [[1-Fv(v)]dvFv(v)dv. 0 -00
1.5.3 Suppose that X is a discrete random variable having the geometric distribution whose probability mass function is p.k/ D p.1????p/k for k D 0;1; : : : :(a) Determine the upper tail probabilities PrfX > kg for k D 0;1; : : : .(b) Evaluate the mean via E[X] D 6k0 PrfX > kg.
1.5.2 Let X1;X2; : : : ;Xn be independent random variables, all exponentially distributed with the same parameter . Determine the distribution function for the minimum Z D minfX1; : : : ;Xng.
1.5.1 Let X1;X2; : : : be independent and identically distributed random variables having the cumulative distribution function F.x/ D PrfX xg. For a fixed number , let N be the first index k for which Xk > . That is, N D 1 if X1 > IN D 2 if X1 and X2 > ; etc. Determine the probability

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