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theory of probability

Questions and Answers of Theory Of Probability

30. Consider a graph with nodes 1, 2,...,n and the n 2 arcs (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
29. 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
28. 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.
27. 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
25. 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 customer then joins a second system
24. 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
23. 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
22. 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
21. 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
20. 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
19. 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
18. After being repaired, a machine functions for an exponential time with rateλ and then fails. Upon failure, a repair process begins. The repair process proceeds sequentially through k distinct
17. 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
16. 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
15. 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
14. 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
13. 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
12. 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
11. 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
10. 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
9. The birth and death process with parameters λn = 0 and μn = μ,n > 0 is called a pure death process. Find Pij (t).
8. 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 rate μ. Set up the Kolmogorov backward
7. 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
6. 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
5. 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
4. 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
3. 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
2. 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
1. 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
34. Let {X(t), −∞ denote the power spectral density of the process.(i) Show that R(w) > = R( > −w). It can be shown that R(s) = 1 2π ∞−∞R(w)e > iws dw(ii) Use the preceding to show
33. Let Y1 and Y2 be independent unit normal random variables and for some constant w set X(t) = Y1 coswt + Y2 sinwt, −∞
32. Let {X(t),−∞
31. Let {N(t),t 0} denote a Poisson process with rate λ and define Y(t) to be the time from t until the next Poisson event.(a) Argue that {Y(t), t 0} is a stationary process.(b) Compute Cov[Y(t),Y(t
30. Let X(t) = N(t + 1) − N(t) where {N(t), t 0} is a Poisson process with rate λ. Compute Cov[X(t),X(t + s)]
29. Let {Z(t), t 0} denote a Brownian bridge process. Show that if Y(t) = (t + 1)Z(t/(t + 1))then {Y(t), t 0} is a standard Brownian motion process.
28. For s
27. Let Y(t) = B(a2t)/a for a > 0. Argue that {Y(t)} is a standard Brownian motion process.
26. Let Y(t) = tB(1/t), t > 0 and Y(0) = 0.(a) What is the distribution of Y(t)?(b) Compare Cov(Y(s),Y(t)).(c) Argue that {Y(t), t 0} is a standard Brownian motion process.
25. Compute the mean and variance of(a) 1 0 t dB(t).(b) 1 0 t2 dB(t).
24. Let {X(t), t 0} be Brownian motion with drift coefficient μ and variance parameter σ2. Suppose that μ > 0. Let x > 0 and define the stopping time T (as in Exercise 21) by?
23. Let X(t) = σB(t) + μt, and define T to be the first time the process{X(t), t 0} hits either A or −B, where A and B are given positive numbers.Use the Martingale stopping theorem and part (c)
22. Let X(t) = σB(t) + μt, and for given positive constants A and B, let p denote the probability that {X(t), t 0} hits A before it hits −B.(a) Define the stopping time T to be the first time the
21. Let {X(t), t 0} be Brownian motion with drift coefficient μ and variance parameter σ2. That is, X(t) = σB(t) + μt Let μ > 0, and for a positive constant x let T = Min{t : X(t) = x}= Mint :
20. Let T = Min{t : B(t) = 2 − 4t}That is, T is the first time that standard Brownian motion hits the line 2 − 4t. Use the Martingale stopping theorem to find E[T ].
19. Show that {Y(t),t 0} is a Martingale when Y(t) = exp{cB(t) − c2t/2}where c is an arbitrary constant. What is E[Y(t)]?An important property of a Martingale is that if you continually observe the
18. Show that {Y(t), t 0} is a Martingale when Y(t) = B2(t) − t What is E[Y(t)]?Hint: First compute E[Y(t)|B(u), 0 u s].
17. Show that standard Brownian motion is a Martingale.
16. If {Y(t), t 0} is a Martingale, show that E[Y(t)] = E[Y(0)]
15. The current price of a stock is 100. Suppose that the logarithm of the price of the stock changes according to a Brownian motion with drift coefficient μ = 2 and variance parameter σ2 = 1. Give
14. The present price of a stock is 100. The price at time 1 will be either 50, 100, or 200. An option to purchase y shares of the stock at time 1 for the (present value) price ky costs cy.(a) If k =
13. Verify the statement made in the remark following Example 10.2.
12. A stock is presently selling at a price of $50 per share. After one time period, its selling price will (in present value dollars) be either $150 or $25. An option to purchase y units of the
11. Consider a process whose value changes every h time units; its new value being its old value multiplied either by the factor eσ√h with probability p = 1 2 (1 + μσ√h), or by the factor
10. Let {X(t), t 0} be a Brownian motion with drift coefficient μ and variance parameter σ2. What is the conditional distribution of X(t) given that X(s) = c when(a) s
9. Let {X(t), t 0} be a Brownian motion with drift coefficient μ and variance parameter σ2. What is the joint density function of X(s) and X(t), s
8. Consider the random walk which in each t time unit either goes up or down the amount √t with respective probabilities p and 1 − p where p = 1 2 (1 + μ√t).(a) Argue that as t → 0 the
7. Compute an expression for Pmax t1st2 B(s) > x
6. Suppose you own one share of a stock whose price changes according to a standard Brownian motion process. Suppose that you purchased the stock at a price b +c, c > 0, and the present price isb.
5. What is P{T1 < T−1 < T2}?
4. Show that P{Ta < ∞} = 1, E[Ta]=∞, a = 0
3. Compute E[B(t1)B(t2)B(t3)] for t1 < t2 < t3.
2. Compute the conditional distribution of B(s) given that B(t1) = A, B(t2) =B, where 0 < t1
1. What is the distribution of B(s) + B(t), s t?
36. Verify Equation (9.36).
35. Prove the combinatorial identityn − 1 i − 1=n i− n i + 1+···± n n, i n(a) by induction on i(b) by a backwards induction argument on i—that is, prove it first for i = n, then
34. For the model of Section 9.7, compute for a k-out-of-n structure (i) the average up time, (ii) the average down time, and (iii) the system failure rate.
33. Let Xi be an exponential random variable with mean 8 + 2i, for i = 1, 2, 3.Use the results of Section 9.6.1 to obtain an upper bound on E[maxXi], and then compare this with the exact result when
32. In Section 9.6.1 show that the expected number of Xi that exceed c∗ is equal to 1.
31. Show that the variance of the lifetime of a k-out-of-n system of components, each of whose lifetimes is exponential with mean θ , is given byθ 2n i=k 1i2
30. Compute the expected system lifetime of a three-out-of-four system when the first two component lifetimes are uniform on (0, 1) and the second two are uniform on (0, 2).
29. Show that the mean lifetime of a parallel system of two components is 1μ1 + μ2+ μ1(μ1 + μ2)μ2+ μ2(μ1 + μ2)μ1 when the first component is exponentially distributed with mean 1/μ1 and
28. Find the mean lifetime of a series system of two components when the component lifetimes are respectively uniform on (0, 1) and uniform on (0, 2). Repeat for a parallel system?
27. Let r(p) = r(p,p,...,p). Show that if r(p0) = p0, then r(p) p for p p0 r(p) p for p p0 Hint: Use Proposition 9.2.
26. Prove Lemma 9.3.Hint: Let x = y + δ. Note that f (t) = tα is a concave function when 0 α 1, and use the fact that for a concave function f (t +h)−f (t) is decreasing in t.
25. We say that ζ is a p-percentile of the distribution F if F(ζ) = p. Show that if ζ is a p-percentile of the IFRA distribution F, then F(x) ¯ e−θx , x ζF(x) ¯ e−θx , x ζwhereθ =
24. Show that if F is IFR, then it is also IFRA, and show by counterexample that the reverse is not true.
23. Show that if each (independent) component of a series system has an IFR distribution, then the system lifetime is itself IFR by(a) showing thatλF (t) =iλi(t)where λF (t) is the failure rate
22. Let X denote the lifetime of an item. Suppose the item has reached the age of t. Let Xt denote its remaining life and define F¯t(a) = P{Xt > a}In words, F¯t(a) is the probability that a t-year
21. Consider the following four structures:
20. Let F be a continuous distribution function. For some positive α, define the distribution function G by G(t) ¯ = (F(t)) ¯ αFind the relationship between λG(t) and λF (t), the respective
19. Let X1,X2,...,Xn denote independent and identically distributed random variables and define the order statistics X(1),...,X(n) by X(i) ≡ ith smallest of X1,...,Xn Show that if the distribution
18. Consider a structure in which the minimal path sets are {1, 2, 3} and{3, 4, 5}.(a) What are the minimal cut sets?(b) If the component lifetimes are independent uniform (0, 1) random variables,
17. Let N be a nonnegative, integer-valued random variable. Show that P{N > 0} (E[N])2 E[N2]and explain how this inequality can be used to derive additional bounds on a reliability
16. Compute the upper and lower bounds of r(p) using both methods for the(a) two-out-of-three system and(b) two-out-of-four system.(c) Compare these bounds with the exact reliability when(i) pi ≡
15. Compute upper and lower bounds of the reliability function (using Method 2)for the systems given in Exercise 4, and compare them with the exact values when pi ≡ 1 2 .
14. Compute the reliability function of the bridge system (see Figure 9.11) by conditioning upon whether or not component 3 is working.
13. Let r(p) be the reliability function. Show that r(p) = pir(1i,p) + (1 − pi)r(0i,p)
12. Give the minimal path sets and the reliability function for the structure in Figure 9.22.
11. Give the reliability function of the structure of Exercise 8.
10. Let ti denote the time of failure of the ith component; let τφ(t) denote the time to failure of the system φ as a function of the vector t = (t1,...,tn). Show that max 1js min i∈Aj ti =
9. Component i is said to be relevant to the system if for some state vector x,φ(1i, x) = 1, φ(0i, x) = 0 Otherwise, it is said to be irrelevant.
8. Give the minimal path sets and the minimal cut sets for the structure given by Figure 9.21.
7. The minimal cut sets are {1, 2, 3}, {2, 3, 4}, and {3, 5}. What are the minimal path sets?
6. The minimal path sets are {1, 2, 4}, {1, 3, 5}, and {5, 6}. Give the minimal cut sets.
5. Find the minimal path and minimal cut sets for:
4. Write the structure function corresponding to the following:(a)
3. For any structure function, we define the dual structure φD byφD(x) = 1 − φ(1 − x)(a) Show that the dual of a parallel (series) system is a series (parallel) system.(b) Show that the dual
2. Show that(a) if φ(0, 0,..., 0) = 0 and φ(1, 1,..., 1) = 1, then min xi φ(x) max xi
1. Prove that, for any structure function, φ,φ(x) = xiφ(1i, x) + (1 − xi)φ(0i, x)where(1i, x) = (x1,...,xi−1, 1,xi+1,...,xn),(0i, x) = (x1,...,xi−1, 0,xi+1,...,xn)
56. Suppose that on each play of the game a gambler either wins 1 with probability p or loses 1 with probability 1 − p. The gambler continues betting until she or he is either winning n or losing

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