Probability And Stochastic Processes 1st Edition Ionut Florescu - Solutions

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2.5 Chris is conducting an experiment to test the mean lifetimes of two sets of electric bulbs labeled A and B. The manufacturer claims that the mean lifetime of bulbs in set A is 200 h, while the mean lifetime of the bulbs in set B is 400 h. The lifetimes for both sets are exponentially
2.4 Customers arrive at a bank according to a Poisson process with an average rate of 6 customers per hour. Each arriving customer is either a man with probability p or a woman with probability 1 2 p. It was found that in the first 2 h, the average number of men who arrived at the bank was 8. What
2.3 Three customers A, B, and C simultaneously arrive at a bank with two tellers on duty. The two tellers were idle when the three customers arrived, and A goes directly to one teller, B goes to the other teller, and C waits until either A or B leaves before she can begin receiving service. If the
2.2 Suppose X(t) is a Gaussian random process with a mean E½XðtÞ 5 0 and autocorrelation function RXXðτÞ 5 e2jτj . Assume that the random variable A is defined as follows: A 5 ðB 0 XðtÞdt where B is a uniformly distributed random variable with values between 1 and 5 and is independent of
2.1 Suppose X(t) is a Gaussian random process with a mean E½XðtÞ 5 0 and autocorrelation function RXXðτÞ 5 e2jτj . Assume that the random variable A is defined as follows: A 5 ð1 0 XðtÞdt Determine the following:a. E½Ab. σ2 A
1.10 Students arrive at the professor’s office for extra help according to a Poisson process with an average rate of four students per hour. The professor does not start the tutorial until at least three students are available. Students who arrive while the tutorial is going on will have to wait
1.9 Joe is a student who is conducting experiments with a series of light bulbs. He started with 10 identical light bulbs each of which has an exponentially distributed lifetime with a mean of 200 h. He wants to know how long it will take until the last bulb burns out (or fails). At noontime he
1.8 Customers arrive at the neighborhood bookstore according to a Poisson process with an average rate of 10 customers per hour. Independently of other customers, each arriving customer buys a book with probability 1/8.a. What is the probability that the bookstore sells no book during a particular
1.7 Cars arrive at a gas station according to a Poisson process at an average rate of 12 cars per hour. The station has only one attendant. If the attendant decides to take a 2-min coffee break when there were no cars at the station, what is the probability that one or more cars will be waiting
1.6 Students arrive for a lab experiment according to a Poisson process with a rate of 12 students per hour. However, the lab attendant opens the door to the lab when at least four students are waiting at the door. What is the probability that the waiting time of the first student to arrive exceeds
1.5 A Girl Scout troop sells cookies from house to house. One of the parents of the girls figured out that the probability that they sell a set of packs of cookies at any house they visit is 0.4, where it is assumed that they sell exactly one set to each house that buys their cookies.a. What is the
1.4 A lady invites 12 people for dinner at her house. Unfortunately, the dining table can only seat six people. Her plan is that if six or fewer guests come, then they will be seated at the table (i.e., they will have a sit-down dinner); otherwise she will set up a buffet-style meal. The
1.3 A sequence of Bernoulli trials consists of choosing components at random from a batch of components. A selected component is classified as either defective or nondefective. A nondefective component is considered to be a success, while a defective component is considered to be a failure. If the
1.2 The probability that a patient recovers from a rare blood disease is 0.3. If 15 people are known to have contracted this disease, find the following probabilities:a. At least 10 survive.b. From 3 to 8 survive.c. Exactly 6 survive.
1.1 A sequence of Bernoulli trials consists of choosing seven components at random from a batch of components. A selected component is classified as either defective or nondefective. A nondefective component is considered to be a success, while a defective component is considered to be a failure.
16.20 Consider the Scott (1987), and the later Chesney and Scott (1989) model which introduced for the first time in finance a so-called mean-reverting volatility process. The model is dSt = μSStdt + eYtStdWt, dYt = α(Y¯ − Yt)dt + σY dZt with the difference between the two papers being that
16.19 Consider the model dSt = μSStdt + #YtSβt dBt dYt = μY Ytdt + ξYtdWt, with β ∈ (0, 1], and correlated Brownian motions with coefficient ρ. Show that, if we take the parameters in the model as μy = α2 and ξ = 2α, μS = 0, we can write the model above as dSt = ytSβt dWt dyt =
16.18 Consider a generalization of the Hull and White (1987) and Wiggins (1987)model:dSt = μSStdt + #YtSβt dBt dYt = μY Ytdt + ξYtdWt, where β ∈ (0, 1] and Bt and Wt are Brownian motions correlated with correlation coefficient ρ. Create an Euler–Milstein approximating scheme for this
16.17 Consider the Hull and White (1987) model dSt = μSStdt + #YtStdBt dYt = μY Ytdt + ξYtdWt.The volatility process is √Yt. Show that the volatility process has moments E[#Yt] = #Y0 e 12μY t− 1 8 ξ2t V[#Yt] = #Y0 2eμY t%1 − e− 1 4 ξ2t&and study what happens with these moments when t
16.16 Give the Euler–Milstein approximation scheme for the following SDE:dSt = μSStdt + σSβt dBt, where β ∈ (0, 1]. Generate 10 paths and plot them for the following parameter values:μS = 0.1, σ = 0.7, β = 1 3, S0 = 100
16.15 Generate and plot 10 Monte Carlo simulations of the process dXt = 10(20 − Xt)dt + 0.7dBt, with X0 = 20. Use the Euler scheme for this problem.
16.14 [Exponential martingale] Suppose θ(t, ω)=(θ1(t, ω), θ2(t, ω),...,θn(t, ω)) is a stochastic process in Rn such that θi(t, ω) ∈ νT for all i. Define the process Zt = e t 0 θ(s,ω)dBs− 1 2 t 0 θ2(s,ω)ds, where Bt is an n-dimensional Brownian motion and all terms in the
16.13 Give an explicit solution for the mean-reverting Ørnstein–Uhlenbeck SDE dXt = α(μ − Xt)dt + σdBt, with X0 = x.
16.12 Show that the stochastic process e t 0 c(s)dBs− 1 2 t 0 c2(s)ds is a martingale for any deterministic function c(t). Does the result change if c(t, ω)is a stochastic process such that the stochastic integral is well defined?
16.11 Solve the following SDEs using the general integrating factor method presented in Example 16.11, on page 505, with X0 = 0:a) dXt = α(β − Xt)dt + σXtdBt,.b) dXt = Xt t dt + σtXtdBt,.c) dXt = Xαt + σXtdBt. For what values of α the solution of the equation explodes (is equal to ∞ or
16.10 Prove by induction that T 0Bk t dBt = Bk+1 Tk + 1 − k 2 T 0Bk−1 t dt.
16.9 If Xt ∼ N(0, σ2t) and Yt = eXt , calculate the pdf of Yt. Calculate E[Yt] and V(Yt). Calculate the transition probability P(Yt = y | Yt0 = y0), and give the density of this transition probability.
16.8 If Xt ∼ N(0, t), calculate the distribution of |Xt|. Calculate E|Xt| and V(|Xt|).
16.7 Let Xt, t ≥ 0, be defined as Xt = {Bt | Bt ≥ 0}, ∀t > 0 That is, the process has the paths of the Brownian motion conditioned by the current value being positive.a) Show that the pdf of Xt is fXt (x)=2fBt (x), ∀x ≥ 0.b) Calculate E[Xt] and V(Xt).c) Is Xt a Gaussian process?d) Is Xt
16.6 Using your choice of software, simulate Bt a standard Brownian motion on the interval t ∈ [0, 2]. Using the simulated paths and the Central Limit Theorem, estimate E[B4 2], E[B8 2], E[B20 2 ].Then simulate each of the processes B4 t , B8 t , and B20 t separately and obtain the previous
16.5 Let Bt be a standard Brownian motion started at 0. Use that for any function f we have:E[f(Bt)] = 1√2πt ∞−∞f(x)e− x2 2t dx, to calculate:E[B2k t ]for some k, an integer. As a hint, you may want to use integration by parts and induction to come up with a formula for E[B2k t ].
16.4 Using a software of your choice, generate points (Y1(iΔt), Y2(iΔt)) solvingdY1(t) = −1 2Y1(t)dt − Y2(t)dBt dY2(t) = −1 2Y2(t)dt + Y1(t)dBt using an Euler approximation. Use Y1(0) = 1, Y2(0) = 0, Δt = 1/1000, i ∈{1, 2,..., 1000}. Plot the resulting pairs of points in a
16.3 Calculate the solution to the following SDE:dXt = α(m − Xt)dt + σdBt with X0 = x.The process satisfying this equation is called the mean-revertingØrnstein–Uhlenbeck process.
16.2 Calculate the solution to the following SDE:dXt = μXtdt + σdBt with X0 = x. The process satisfying this equation is called the Ørnstein–Uhlenbeck process.
16.1 Calculate t 0sdBs.To this end, apply Ito lemma to ˆ Xt = Bt and f(t, x) = tx.
15.17 Repeat the previous problem but now generate a path using the process St = 100eμt+σBt ,
15.16 Generate one path of the following process:Xt = 0.01t + 0.3Bt, for t ∈ [0, 1] and sampled with frequency Δt = 1/252. Let the observations be denoted with x0, x1 ...,x252. Estimate the parameters μ and σ of the process Xt = μt + σBt, using these observations. Compare the estimates with
15.15 Generate and plot the following processes. Use the time interval [0, 1] and generate the sampled values with frequency Δt = 1/252.a) A standard Brownian motion started from 0.b) A Brownian motion with drift μ = 0.01 and volatility σ = 0.3c) The process St = 100eμt+σBt , with S0 = 100 and
15.14 Let Bt be a standard Brownian motion. Define Nt = min 0≤s≤t Bs.Calculate the pdf of Nt. Calculate its expected value.
15.13 [Brownian bridge] Let Bt be a Brownian motion started from 0. Consider the process Bt conditional on B1 = 0. That is, consider the process Bt | B1 = 0.a) Show that this process is a Gaussian process.b) Calculate for s
15.12 Suppose we observe the trajectory of a particle at time t = 1 and we see that the position is Xt = 4. We need to estimate the distribution of the particle at time s = 1/2. To do this, we assume that the position at time t is given by a standard Brownian motion. Calculate the distribution of
15.11 If Bt is a Brownian motion on R, show that 1 t B2 t is distributed as a chisquared random variable.
15.10 If Bt is a Brownian motion and Xt = eσBt , for some σ > 0, calculate the pdf of Xt. Calculate E[Xt] and V(Xt). Calculate the transition probability P(Xt ≤ y | Xt0 = y0), and give the density of this transition probability. For this part, use the functions in Definition 15.2.
15.9 If Bt is a Brownian motion, calculate the distribution of |Bt|. Calculate E|Bt|and V(|Bt|).
15.8 Let errf(x) denote the regular error functions, that is, errf(x) = 2√π x 0e−t2 dt.Show that erf(x)=1 − e−x2√π! 1 x − 1 2x3 + ... ".Furthermore, show that limε→0 2ε!1 − Φ! δσ√ε"" = 0 for all σ > 0 and δ > 0. Here, φ(x) denotes the cdf of a N(0, 1) random variable.
15.7 Let Xt, t ≥ 0 be defined as Xt = {Bt | Bt ≥ 0}, ∀t > 0, where Bt is a Brownian motion. That is, the process has the paths of the Brownian motion conditioned by being positive at t.a) Show that the pdf of Xt is fXt (x)=2fBt (x), ∀x ≥ 0.
15.6 Using your choice of software, simulate Bt a standard Brownian motion on the interval t ∈ [0, 2]. Using the simulated paths and the Central Limit Theorem, estimate E[B4 2], E[B8 2], E[B20 2 ].Then simulate each of the processes B4 t , B8 t , B20 t separately and obtain the previous
15.5 Let Bt a standard Brownian motion on R.a) Show that for all f such that the integral exists we have E[f(Bt)] = 1√2πt ∞−∞f(x)e− x2 2t dx.b) Apply the previous part to f(x) = x2k and find a recurrence relation between E[B2k t ] and E[B2(k−1)t ].c) Use the previous part and
15.4 [Ornstein–Uhlenbeck process] Consider the process Xt = e−t Be2t , where Bt is a standard Brownian motion and the time corresponding to t in Xt is e2t for the Brownian motion.a) Calculate E[Xt] and V(Xt).b) Show that Xt has a normal distribution.c) Show that Xt and X−t have the same
15.3 Let Bt a standard Brownian motion. Show that Xt = 3B t 9 is also a standard Brownian motion.
15.2 Calculate the characteristic function of a Gaussian vector (see Theorem 15.4 on page 469).
15.1 Show that the standard Brownian motion is a martingale. Show that it is a Markov process.
14.16 [Doob’s inequalities] For a stochastic process X = {Xn}n with a filtrationℱ = {ℱn}n, we define the maximum process X¯ n(ω) = max 0≤i≤n Xi(ω).a) Suppose the original process X = {Xn}n is a submartingale. Show that P(X¯ n ≥ λ) ≤1λE )Xn1X¯n≥λ1≤1λE )Xn1X¯n≥0 1= E
14.15 Recall Section 14.5.3.1 on page 454 and the definition of the hitting time T given there. Let M(θ) be the moment generating function of the Xi’s, φ(θ) =log(M(θ)), and Xθn = eθSn−nφ(θ).a) Calculate M(θ) and φ(θ) for θ ∈ R, and show that φ(θ) ≥ 0 for all θ > 0.b) For θ >
14.14 Suppose that X1,...,Xn,... are i.i.d. random variables. Construct the partial sums process:Sn = n i=1 Xi, and the standard filtration ℱn = σ(X1,...,Xn). Suppose that there exists an M > 0 such that the moment generating function for the Xi’s is finite on (−M, M), that is,
14.13 Consider a branching process with m the expected offspring size. We have seen that the process Xn/mn is a martingale where Xn is the size of generation n.Show that if m > 1, then Xn → ∞ almost surely.
14.12 Consider a simple random walk with p = 1/2 started from S0 = 0, and let Tα = inf{n ∈ N | Sn = α or Sn = −α}, the first time the random walk exits from (−α, α), or equivalently first hitting time of(−∞, α]∪[α,∞). Using the methodology presented in the examples, find an
14.11 Suppose that random variables Xn are such that the distribution of Xn+1 given Xn is Binomial(10, Xn 10 ).
14.10 [Wright–Fisher population model] In a population we have two types of individuals, type A and type B. Type A is purely dominant and type B carries a particular type of recessive allele. Let Xn be the proportion of population of type A in generation n. Suppose the population is always
14.9 [Polya’s urn] Suppose an urn contains one white ball and one green ball. At time 1, a ball is selected randomly and a ball of the same color is added to the urn so that there are now three balls in the urn. Then at time 2, a ball from the urn is selected randomly and a new ball of the same
14.8 Consider De Mere’s martingale in the previous problem. To eliminate such strategies, most casinos impose a maximum amount that can be bet at any one game
14.7 [De Mere’s martingale] We have a fair coin p = 1/2 and a fair game i.e., twice the amount gambled if the toss is won, and lose all amount if the toss is lost.Suppose we adopt a doubling strategy, that is, we start with x dollars, if we lose, we next bet 2x, and so on. If we lose n bets, the
14.6 For a simple random walk Sn, which of the following are stopping times?a) T1 = min{n ≥ 3 | Sn = Sn−3 + 3},b) T2 = min{n ≥ 3 | Sn = Sn−3 + 2},c) T3 = T1 + 3,d) T4 = T2 ∧ T3,e) T5 = T1 ∨ 5.
14.5 Let X1, X2,...,Xn be i.i.d. with mean 0 and variance σ2 finite. Suppose T is a stopping time for the standard filtration generated by the Xi’s with a finite expectation. Show that Var T i=1 Xi= σ2E[T].
14.4 Let Xn be in L1 and ℱn = σ(X1, X2,...,Xn) its standard filtration. Suppose that we have E[Xn+1] = aXn + bXn−1, for somea, b ∈ (0, 1) with a + b = 1. Find α so that the process Zn = αXn + Xn−1 is a martingale with respect to ℱn.
14.3 [Not a stopping time] We saw the first entry defined as an infimum is a stoping time. How about the supremum? Let Xt be a martingale and let S = sup{n | n ≤ 100, Xn ∈ B}for some interval B in R with the convention sup(∅)=0. Give an argument to show that S is not a stopping time.
14.2 Let X = {Xn}n be a martingale with X0 = 0 and let ℱn be its standard filtration. Show that E[Xn]2 = n i=1 E[(Xi − Xi−1)2].
14.1 Let X = {Xn}n a stochastic process so that Xi’s are squared integrable and let ℱn be its standard filtration. Suppose that Zn = n i=1 Xn is a martingale. Show that E[XiXj ]=0 for all i = j.
13.20 Refer to the previous problem and assume that in fact the number of traders submitting orders is finite of size M. Thus, instead of growing at a constant rate, it makes more sense for the resulting chain to have qi,i+1 = (M − i)λ, for i ∈ {0, 1,...,K − 1}
13.19 Suppose orders are coming to a market exchange according to a Poisson process with rate λ from high-frequency traders. However, the exchange can only process K orders at the same time, so an order sent when the system is processing K orders is lost. Furthermore, assume that the order
13.18 Consider an M/M/s queue with s servers, where the customers arrive with rate λ and each server processes with rate μ independently of others. Show that the number of customers in the queue is a birth and death process. Show that its stationary distribution can be calculated using?
13.17 Consider a discrete-time Markov chain with transition times in integers, with the following transition probability matrix:
13.16 Let Xt denote a Markov process with two states 1 and 2, and let q12 = 1 and q21 = 1/2. Show that the Markov process is reversible. Calculate the limiting probabilities P1 and P2.
13.15 Consider a birth and death model with immigration as in the previous problem. However, assume that once the population has reached a size N immigration is forbidden. Set this process as a birth–death model. Suppose N = 6, λ = θ = 2, andμ = 4. Calculate the proportion of time that
13.14 Consider a population as in the previous problem, but in addition new citizens immigrate according to a Poisson process with a rate θ. This process is called a birth and death process with immigration. Since each immigrant arrives alone, show that this process is equivalent to a
13.13 Consider a population in which each individual independently gives birth at an exponential rate λ and dies at an exponential rate μ. What kind of process is this?What are its parameters? Find E[Xt | X0 = n].
13.12 Everyone has seen this situation. You chose one line and see that the other lines next to the line you chose are moving faster. Yet other times, lines are combined into one and the next customer in line can go to any server. In this exercise, we will compare the two situations using a very
13.11 Suppose a shop has three machines and a repairman. Machines are breaking down independently with rate λ = 3 per week and the rate at which the repairman fixes any given machine is constant μ = 1 per day. Construct a Markov process Xt which counts the number of machines broken down at time
13.10 Consider an M/M/1/c queue. This is a queuing process where arrivals are according to a Poisson process with rate λ the service is exponentially distributed with rate μ, there is one server, and the queue capacity isc. Write this queue as a Markov chain in continuous time. Calculate the
13.9 Consider a Markov process with state space S = {0, 1, 2,... } and transition rates qi,i+1 = ai, qi,0 =b, and qi,k = 0 in all other cases. Show the following:a) When a > 1, this process is not regular.b) Find the stationary distribution when a ≤ 1 and b > 0.c) Show that the stationary
13.8 Consider the Yule process presented in Example 13.6 on page 425. Calculate the transition probabilities Pii(t) and P37(0.5).
13.7 Use the backward Kolmogorov equations to find to find P01 and P00 in Example 13.3.
13.6 Prove Lemma 13.15.
13.5 Prove Proposition 13.11 on page 418.
13.4 Show that the continuous-time Markov chain defined in Example 13.2 on page 417 is nonregular.
13.3 Suppose a semi-Markov process has two states S = {1, 2}. Assume that the distributions of times spent in each state are the cdfs of the following distributions:F11(t) ∼ Uniform[0, 1]; F12(t) ∼ Uniform[0, 3];F21(t) ∼ Uniform[1, 3]; F22(t) ∼ Uniform[0, 2].Calculate the transition rates
13.2 Consider a sequence of Bernoulli(1/2) random variables X0, X1,...,Xn,....Construct a process Yn = Xn − Xn−1.1. Does the process Yn have the Markov property?2. Define the two-dimensional process Zn = (Yn, Xn−1). Show that this twodimensional process has the Markov property.
13.1 Consider a sequence of Bernoulli(1/2) random variables X0, X1,...,Xn,....Construct a process Yn = Xn−1 + 2Xn + 3Xn+1.1. Calculate P(Y0 = 1, Y1 = 3, Y2 = 2) and P(Y1 = 3, Y2 = 2).2. Calculate P(Y2 = 2 | Y0 = 1, Y1 = 3) and P(Y2 = 2 | Y1 = 3).3. Does the process Yn have the Markov property?
12.21 Suppose the color of rats is determined by three different alleles B, G, and b.Suppose B and G are codominant alleles. There are four different types of individual rat colors even though there are six genotypes:
12.20 Refer to the genetic transition presented in the previous problem. Suppose that I know that my parents have brown eyes (genotype Bg) and blue eyes (genotype bg). I know that I have brown eyes and my wife has blue eyes. Calculate the probability that my son has blue eyes.
12.19 Assume it is known that the color of eyes of humans is determined by a particular gene. We assume that this gene has three alleles B,b, and g, where B is dominant and both b and g are recessive. A person with genotype containing the dominant allele B has brown eyes. A person with both
12.18 In this problem, we look at the model presented in Example 12.11 and the resulting transition probability presented in Table 12.6. Identify the transient and recurrent states of the Markov chain. Reorder the states so that the recurrent states are ordered first, and write down the resulting
12.17 We refer again to the Example 12.10 on page 401. Suppose that the individuals we choose for crossbreeding are always coming from a sample which contains 25% individuals with genotype BB, 50% individuals with genotype Bb, and 25% individuals with genotype bb. Calculate offspring transition
12.1. Calculate the stationary distribution if it exists. Suppose we start with a single fly and in generation 1000 we have 50 individuals. What proportion of these is expected to have white eyes.b) Repeat parta) for the Markov chain described by the transition probabilities in Table 12.2.c) Repeat
12.16 In this problem, we will analyze the model presented in Example 12.10 on page 401. Read and understand this example first.a) Consider a Markov chain described by the transition probabilities in Table
12.15 A baseball player is very emotional. Every time he gets a hit, his confidence grows and the probability of making a hit at the next try increases. Similarly, if he strikes out or fails to make a hit, his confidence drops and his probability of getting a ?
12.14 In a network queue transmission packets go through two consecutive processes. Router A processes the packets in a time distributed with a geometric distribution with mean 10 ms. Router B routs the packets in a time distributed as a Geometric(20) random variable (also in milliseconds). All the
12.13 Suppose we have an asymmetric random walk on integers with Pi,i+1 = p and Pi,i−1 = 1 − p for all i ≥ 1. For i = 0, we have P0,1 = p and P0,0 = 1 − p.Classify the states of the Markov chain and find its stationary distribution. Discuss the existence of the stationary distribution
12.12 Consider an irreducible Markov chain. Show that, if it is periodic, then Pii =0 for all its states i. Is the reciprocal implication true, that is, if Pii = 0 for all i, is the chain periodic? Give a proof or counterexample.
12.11 Consider a random walk with n + 1 states. The transition probability from state i to state j is Pij =⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩1 − p if i = j = 0 p if j = i + 1, and i = 0,...,n − 1 p if i = j = n 1 − p if j = i − 1, and j = 1,...,n 0 otherwise.Draw the states and

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Probability And Stochastic Processes 1st Edition Ionut Florescu - Solutions

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