Markov Processes For Stochastic Modeling 1st Edition Oliver Ibe - Solutions

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26 An ordinary deck of 52 cards is dealt, 13 each, to four players at random. What is the probability that each player receives 13 cards of the same suit?
25 What is the coefficient of x'y' in the expansion of (2x-y+3)
24 What is the coefficient of xyz2 in the expansion of (2x - y + 3z)??
23 Using Theorem 2.6, expand (x + y + z).
22 A history professor who teaches three sections of the same course every semester decides to make several tests and use them for the next 10 years (20 semesters) as final exams. The professor has two policies: (1) not to give the same test to more than one class in a semester, and (2) not to
21 There are 12 nuts and 12 bolts in a box. If the contents of the box are divided between two handymen, what is the probability that each handyman will get six nuts and six bolts?
20 Poker hands are classified into the following 10 nonoverlapping categories in increasing order of likelihood. Calculate the probability of the occur- rence of each class separately. Recall that a poker hand consists of five cards selected randomly from an ordinary deck of 52 cards. Royal flush:
19 Suppose that 12 married couples take part in a contest. If 12 persons each win a prize, what is the probability that from every couple one of them 24 is a winner? Assume that all of the (22) possible sets of winners are equally probable. 12
18 A fair die is tossed six times. What is the probability of getting exactly two 6's?
17 Find the values of 2" (") and ("). 1=0
16 A lake contains 200 trout; 50 of them are caught randomly, tagged, and returned. If, again, we catch 50 trout at random, what is the probability of getting exactly five tagged trout?
15 From a faculty of six professors, six associate professors, 10 assistant professors, and 12 instructors, a committee of six is formed randomly. What is the probability that (a) there are exactly two professors on the committee; (b) all committee members are of the same rank?
14 If five numbers are selected at random from the set {1, 2, 3, ..., 20), what is the probability that their minimum is larger than 5?
13 A fair coin is tossed 10 times. What is the probability of (a) five heads; (b) at least five heads?
12 A team consisting of three boys and four girls must be formed from a group of nine boys and eight girls. If two of the girls are feuding and refuse to play on the same team, how many possibilities do we have?
11 Find the coefficient of x3y4 in the expansion of (2x - 4y).
10 Find the coefficient of x' in the expansion of (2 + x)12.
9 In front of Jeff's office there is a parking lot with 13 parking spots in a row. When cars arrive at this lot, they park randomly at one of the empty spots. Jeff parks his car in the only empty spot that is left. Then he goes to his office. On his return he finds that there are seven empty spots.
8 Lili has 20 friends. Among them are Kevin and Gerry, who are husband and wife. Lili wants to invite six of her friends to her birthday party. If neither Kevin nor Gerry will go to a party without the other, how many choices does Lili have?
7 Ann puts at most one piece of fruit in her child's lunch bag everyday. If she has only three oranges and two apples for the next eight lunches of her child, in how many ways can she do this?
6 Judy puts one piece of fruit in her child's lunch bag everyday. If she has three oranges and two apples for the next five days, in how many ways can she do this?
5 A random sample of n elements is taken from a population of size N without replacement. What is the probability that a fixed element of the population is included? Simplify your answer.
4 From an ordinary deck of 52 cards, five are drawn randomly. What is the probability of drawing exactly three face cards?
3 A panel consists of 20 men and 25 women. How many choices do we have to obtain a jury of six men and six women from this panel?
2 Each of the 50 U.S. states has two senators. In how many ways may a majority be achieved in the U.S. Senate? Ignore the possibility of absence or abstention. Assume that all senators are present and voting.
1 Jim has 20 friends. If he decides to invite six of them to his birthday party, how many choices does he have?
19 The coefficient of the quadratic equation ax+bx+c= 0 are determined by tossing a fair die three times (the first outcome isa, the second oneb, and the third one c). Find the probability that the equation has no real roots.
18 A bus traveling from Baltimore to New York breaks down at a random location. What is the probability that the breakdown occurred after passing through Philadelphia? The distances from New York and Philadelphia to Baltimore are, respectively, 199 and 96 miles.
17 Suppose that each day the price of a stock moves up 1/8 of a point, moves down 1/8 of a point, or remains unchanged. For i 1, let U; and D be the events that the price of the stock moves up and down on the ith trading day, respectively. In terms of U's and D's find an expression for the event
16 A number is selected at random from the set (1, 2, 3, ..., 150}. What is the probability that it is relatively prime to 150? See Exercise 21 of Section 1.4 for the definition of relatively prime numbers.
15 A number is selected at random from the set of natural numbers (1, 2, 3, ... 1000}. What is the probability that it is not diyisible by 4, 7, or 9?
14 Suppose that in a certain town the number of people with blood type O and blood type A are approximately the same. The number of people with blood type B is 1/10 of those with blood type A and twice the number of those with blood type AB. Find the probability that the next baby born in the town
13 A bookstore receives six boxes of books per month on six random days of each month. Suppose that two of those boxes are from one publisher, two from another publisher, and the remaining two from a third publisher. Define a sample space for the possible orders in which the boxes are received in a
12 Let A and B b two events. The event (AB) U (BA) is called the symmetric difference of A and B and is denoted by AA B. Clearly, AAB is the event that exactly one of the two events A and B occurs. Show that P(AAB) = P(A) + P(B)-2P(AB).
11 Let A and B be two events. Suppose that P(A), P(B), and P(AB) are given. What is the probability that neither A nor B will occur?
10 Answer the following question, from the "Ask Marilyn" column of Parade Magazine, March 3, 1996. My dad heard this story on the radio. At Duke University, two students had received A's in chemistry all semester. But on the night before the final exam, they were partying in another state and
9 Suppose that 40% of the people in a community drink or serve white wine, 50% drink or serve red wine, and 70% drink or serve red or white wine. What percentage of the people in this community drink or serve both red and white wine?
8 Let A, B, and C be three events. Show that P(AUBUC) P(A) + P(B) + P(C) if and only if P(AB) = P(AC) = P(BC) = 0.
7 Let A, B, and C be three events. Prove that P(AUBUC) P(A) + P(B) + P(C).
6 The number of patients now in a hospital is 63. Of these 37 are male and 20 are for surgery. If among those who are for surgery 12 are male, how many of the 63 patients are neither male nor for surgery?
5 A coin is tossed until, for the first time, the same result appears twice in succession. Define a sample space for this experiment.
4 The following relations are not always true. In each case give an example to refute them.(a) P(AUB) = P(A) + P(B). (b) P(AB) = P(A)P(B).
3 In a certain experiment, whenever the event A occurs, the event B also occurs. Which of the following statements is true and why? (a) If we know that A has not occurred, we can be sure that B has not occurred as well. (b) If we know that B has not occurred, we can be sure that A has not occurred
2 Define a sample space for the experiment of choosing two distinct sets randomly from the set of all subsets of A = {1, 2}. Describe the fol- lowing events: The intersection of these sets is empty; these two sets are complements of each other; one set contains more elements than the other.
1 The number of minutes it takes for a certain animal to react to a certain stimulus is a random number between 2 and 4.3. Find the probability that the reaction time of such an animal to this stimulus is no longer than 3.25 minutes.
14 Let A be the set of rational numbers in (0, 1). Since A is countable it can be written as a sequence [i.e., A = (r: n = 1, 2, 3, ...}]. Prove that for any > 0, A can be covered by a sequence of open balls whose total length is less than . That is, Ve > 0, there exists a sequence of open
13 Show that the result of Exercise 8 is not true for an infinite number of events. That is, show that if [E,:0 <
12 Let (A1, A2, A3, ...) be a sequence of events. Prove that if the series = P(A) converges, then P (UA) = 0. This is called the Borel-Cantelli lemma. It says that if P(A)
11 Suppose that a point is randomly selected from the interval (0, 1). Using the definition in Section 1.7, show that all numerals are equally likely to appear as the nth digit of the decimal representation of the selected point.
10 A point is selected at random from the interval (0, 1). What is the prob- ability that it is rational? What is the probability that it is irrational?
9 = (a) Prove that (1/2-1/2n, 1/2 + 1/2n) = (1/2). == (b) Using part (a), show that the probability of selecting 1/2 in a random selection of a point from (0, 1) is 0.
8 Let A1, A2, A be n events. Show that if P(A) = P(A2) = P(A) = 1, then P(AA2A,) = 1.
7 Is it possible to define a probability on a countably infinite sample space so that the outcomes are equally probable?
6 Suppose that a point is randomly selected from the interval (0, 1). Using the definition in Section 1.7, show that all numerals are equally likely to appear as the first digit of the decimal representation of the selected point.
5 A point is selected at random from the interval (0, 2000). What is the probability that it is an integer?
4 Let A and B be two events. Show that if P(A) = 1 and P(B) = 1, then P(AB)=1.
3 Which of the following statements are true? If a statement is true, prove it. If it is false, give a counterexample. (a) If A is an event with probability 1, then A is the sample space. (b) If B is an event with probability 0, then B = 0.
2 Past experience shows that every new book by a certain publisher captures randomly between 4 and 12% of the market. What is the probability that the next book by this publisher captures at most 6.35% of the market?
1 A bus arrives at a station every day at a random time between 1:00 PM. and 1:30 PM. What is the probability that a person arriving at this station at 1:00 PM. will have to wait at least 10 minutes?
Consider the geometric Brownian motion Y(t), which is used to model stock prices. It is characterized by the drift rate μ > 0 and the standard deviation parameter σ that is usually referred to as the volatility in the field of finance. The stochastic differential equation (SDE) is given by dY(t)
Simulate a standard Brownian motion by repeatedly generating independent normal random variables with mean 0 and standard deviation√(1/N), where N is the number of experiments. Specifically, assume an interval from 0 to 1 that is divided into N equally spaced points. Let the points T1, T2,...,TN
Consider the following acceptance probability for the Metropolis-Hastings algorithm:α(x, y) = π(y)π(x) + π(y)Show that this definition of α(x, y) produces a reversible Markov chain and has the stationary distribution π if the transition probability q(x|y) is symmetric.
Consider a 50 × 50 square array of spins. Initially assign all sites the spin value of +1 and select one spin randomly. The simulation involves flipping a random lattice point and determining if the net energy decreases. If the energy decreases because of the flip, then the new state is allowed,
A self-avoiding random walk (SAW) is a special random walk along an m-dimensional lattice such that no vertex (or site) is visited more than once, Hayes (1998). In this way the trajectory never intersects itself.Self-avoiding random walks arise in modeling physical processes like the Markov Chain
Write a simulation program that generates random samples from the standard normal distribution, N(0, 1). Generate at least 2000 samples and record the number of samples that fall into each of the following intervals:(−∞,−2), (−2,−1), (−1, 0), (0, 1), (1, 2), (2,∞). Use these numbers
Consider the function f(x) =⎧⎨⎩3x3 2 −1 ≤ x ≤ 1 0 otherwise Suggest a probability density function π(x), where −1 ≤ x ≤ 1, and a function h(x) such that we can obtain the following Monte Carlo integration: 1−1 f(x)dx = 1−1 h(x)π(x)dx = Eπ[h(x)]∼= 1 nn k=1 h(xk)
Consider the PDF given by fX(x) =⎧⎨⎩3x2 2 −1 ≤ x ≤ 1 0 otherwise If we define Y = FX(x), use the inverse transform method to obtain x = F −1 X (Y).
Consider a random variable X whose PDF is given by fX(x) = 1/(1+x)2, where x > 0. If we define Y = FX(x), obtain x = F −1 X (Y ).
Consider the Weibull random variable X with the CDF FX(x) = 1 −exp{−[(x − x0)/a]c} where x>x0 anda, c > 0 are parameters. Obtain x = F −1 X (Y ), where Y = FX(x).
One important parameter in stationary point processes is the autointensity function. The autointensity function, hNN(t, v), of the process N(t) is the conditional probability that a point occurs at time t + v, given that a point occurs at time t. Specifically, hNN(t, v) = P[dN(t + v) = 1|dN(t) =
Passengers arrive at a train station according to a Poisson process with a rate of 25 customers per hour. It has been found that 60% of the passengers are female. What is the probability that no male customer arrives at the station over a 15-minute period?
A restaurant has two entrances, A and B. Customers arrive at the restaurant through entrance A according to a Poisson process with a rate of five customers per hour, and customers arrive through entrance B according to a Poission process with a rate of seven customers per hour.a. What is the
Let {X1, X2,...} be a sequence of independent and identically distributed random variables with PDF fX(x), and let N be an integer-valued random variable with PMF pN(n), where N and the Xi are independent. Consider a process in which events occur at times X1, X1 + X2,... , X1 + X2+···+ XN .a.
A hard-core point process is produced from a Poisson process of rate λby deleting any point within distance v0 of another point, regardless of whether that point has itself already been deleted. Prove that the rate of the hard-core process is λ exp(−λπv2 0).
Two classes of customers arrive at Paul’s barber shop: class 1 and class 2. Class 1 customers arrive according to a Poisson process with rate λ1 customers per hour, and class 2 customers arrive according to a Poisson process with rate λ2 customers per hour.a. What is the probability that no
Assume that the definition of the clique potential for Problem 12.3 is expanded to include horizontal pair neighbors, vertical pair neighbors, and diagonal pair neighbors. Obtain the following ratios:a. P[x2]/P[x1]b. P[x3]/P[x1]
For the three 4 × 4 binary image data shown in Figure 12.9 that correspond to the Gibbs random field realizations denoted by X = x1, X = x2, and X = x3, respectively, assume that the clique potential is defined as follows:Vc(x) =β if both horizontal or vertical pair elements are identical−β
Consider the three 4 × 4 binary image data shown in Figure 12.9 that correspond to the Gibbs random field realizations denoted by X = x1, X = x2, and X = x3, respectively. Let the clique potential be defined as follows:Vc(x) =β if both elements in the horizontal pair clique are identical−β
Show that the Markov chain {Xn, n = 0, 1,...,N} is a Markov random field.Hint: show that P[X] = P[X0, X1,...,XN] can be expressed in the form P[X = x] = Z−1 exp{−βU(x)} and identify the parameters Z and V(x) =βU(x).
Consider the HMM shown in Figure 11.19 that has three hidden states 1, 2, and 3, and emits two output symbols: U and V. When it is in state 1, it is equally likely to emit either symbol. When it is in state 2, the probability that it emits the symbol U is 0.1, and the probability that it emits the
Consider an HMM with two states 1 and 2 and that emits two symbols:A and B. The state-transition diagram is shown in Figure 11.18.a. Use the Viterbi algorithm to obtain the most likely state sequence that produced the observation sequence {ABBAB}.b. Estimate the probability that the sequence
The price of a certain stock is fluctuating among $10, $20, and $30 from month to month. Market analysis indicates that given that the stock is at$10 in the current month, then in the following month it will be at $10 with probability 4/5 and at $20 with probability 1/5. Similarly, given that the
A recent college graduate is presented with N job offers, one after another.After looking at an offer, she must either accept it and thus terminate the process or reject it. A rejected offer is lost forever. The only information she has at any time is the relative rank of the current offer compared
Let {B(t), t ≥ 0} be a standard Brownian motion, and define the process Y(t) = e−t B(e2t), t ≥ 0; that is, {Y(t),t ≥ 0} is the Ornstein-Uhlenbeck process.a. Show that Y(t) is a Gaussian process.b. Find the covariance function of the process.
Let the process {X(t), t ≥ 0} be defined by X(t) = B2(t) − t, where {B(t), t ≥ 0} is a standard Brownian motion.a. What is E[X(t)]?b. Show that {X(t), t ≥ 0} is a martingale.Hint: Start by computing E[X(t)|B(v), 0 ≤ v ≤ s].
What is the mean value of the first passage time of the reflected Brownian motion {|B(t)|,t ≥ 0} with respect to a positive level x where B(t) is the standard Brownian motion? Determine the CDF of |B(t)|.
Consider the Brownian motion with drift Y(t) = μt + σB(t) + x where Y(0) = x and b
Let Y(t) = t 0B(u)du; where {B(t), t ≥ 0} is the standard Brownian motion.Finda. E[Y(t)]b. E[Y2(t)]c. The conditional distribution of Y(t), given that B(t) = x
Let T = min{t:B(t) = 5 − 3t}. Use the martingale stopping theorem to find E[T ].
Let {X(t), t ≥ 0} be a Brownian motion with drift rate μ and variance parameter σ2. What is the conditional distribution of X(t) given that X(u) =b, u < t?296 Markov Processes for Stochastic Modeling
Suppose X(t) is a standard Brownian motion and Y(t) = tX(1/t). Show that Y(t) is a standard Brownian motion.
Assume that X and Y are independent random variables such that X ∼N(0, σ2) and Y ∼ N(0, σ2). Consider the random variables U = (X + Y )/2 and V = (X − Y )/2. Show that U and V are independent with U ∼N(0, σ2/2) and V ∼ N(0, σ2/2).
Consider a continuous-time random walk {X(t):t ≥ 0} in which the jump size, , is normally distributed with mean μ and variance σ2, and the waiting time, T , is exponentially distributed with mean 1/λ. Obtain the master equation, P(x,t), which is the probability that the position of the walker
Consider a coordinated random walk with stay. That is, a walker can move to the right, to the left, or not move at all. Given that the move in the current step is to the right, then in the next step it will move to the right again with probabilitya, to the left with probabilityb, and remain in the
Consider a cash management scheme in which a company needs to maintain the available cash to be no more than $K. Whenever the cash level reaches K, the company buys treasury bills and reduces the cash level to x.Whenever the cash level reaches 0, the company sells enough treasury bills to bring the
Consider an asymmetric random walk that takes a step to the right with probability p and a step to the left with probability q = 1 − p. Assume that there are two absorbing barriers, a andb, and that the walk starts at the point k, where b
Let N denote the number of times that an asymmetric random walk that takes a step to the right with probability p and a step to the left with probability q = 1 − p revisits its starting point. Show that the PMF of N is given by pN(n) = P[N = n] = β(1 − β)n n = 0, 1, 2,...where β = |p −
Consider the random walk Sn = X1 + X2 +···+ Xn, where the Xi are independent and identically distributed Bernoulli random variables that take on the value 1 with probability p = 0.6 and the value −1 with probability q = 1 − p = 0.4.a. Find the probability P[S8 = 0].b. What value of p
Consider a single-server discrete-time queueing system that operates in the following manner. Let Xn denote the number of customers in the system at time n ∈ {0, 1, 2,...}. If a customer is receiving service in time n, then the probability that he finishes receiving service before time n + 1 is
Chris has $20 and Dana has $30. They decide to play a game in which each pledges $1 and flips a fair coin. If both coins come up on the same side, Chris wins the $2, and if they come up on different sides, Dana wins the $2. The game ends when either of them has all the money. What is the

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Markov Processes For Stochastic Modeling 1st Edition Oliver Ibe - Solutions

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