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theory of probability

Questions and Answers of Theory Of Probability

A population consists of m families. Let Xj denote the size of family j, and suppose that X1, . . . ,Xm are independent random variables having the common probability mass functionwith mean μ =
Consider n independent trials in which each trial results in one of the outcomes 1, . . . , k with respective probabilities p1, . . . , pk,ki=1 pi = 1.Suppose further that n > k, and that we are
An automobile insurance company classifies each of its policyholders as being of one of the types i = 1, . . . , k. It supposes that the numbers of accidents that a type i policyholder has in
1. If X and Y are both discrete, show thatx pX|Y(x|y) = 1 for all y such that pY(y) > 0.
2. Let X1 and X2 be independent geometric random variables having the same parameter p. Guess the value of P{X1 = i|X1 + X2 = n}Hint: Suppose a coin having probability p of coming up heads is
3. The joint probability mass function of X and Y, p(x, y), is given byCompute E[X|Y = i] for i = 1, 2, 3. p(1, 1), p(2, 1) = p(3, 1) =, p(1,2) = p(2,2)=0, p(3,2)=, p(1,3)=0, p(2,3)= p(3,3)=
4. In Exercise 3, are the random variables X and Y independent?
23. A coin having probability p of coming up heads is successively flipped until two of the most recent three flips are heads. Let N denote the number of flips. (Note that if the first two flips are
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)
53. Consider a model in which the interarrival times have an arbitrary distribution F, and there are k servers each having service distribution G. What condition on F and G do you think would be
52. Consider a system where the interarrival times have an arbitrary distribution F, and there is a single server whose service distribution is G. Let Dn denote the amount of time the nth customer
51. Verify the formula for the distribution of W∗Q given for the G/M/k model.
50. In the M/M/k system,(a) what is the probability that a customer will have to wait in queue?(b) determine L and W.
49. In the Erlang loss system suppose the Poisson arrival rate is λ = 2, and suppose there are three servers, each of whom has a service distribution that is uniformly distributed over (0, 2). What
48. Verify the formula given for the Pi of the M/M/k.
47. Verify Erlang’s loss formula, Equation (8.60), when k = 1.
46. In the G/M/1 model if G is exponential with rate λ show that β = λ/μ.
45. Calculate explicitly (not in terms of limiting probabilities) the average time a customer spends in the system in Exercise 24.
44. Consider the priority queueing model of Section 8.6.2 but now suppose that if a type 2 customer is being served when a type 1 arrives then the type 2 customer is bumped out of service. This is
43. In a two-class priority queueing model suppose that a cost of Ci per unit time is incurred for each type i customer that waits in queue, i = 1, 2. Show that type 1 customers should be given
42. In the two-class priority queueing model of Section 8.6.2, what is WQ?Show that WQ is less than it would be under FIFO if E[S1] < E[S2] and greater than under FIFO if E[S1] > E[S2].
41. Carloads of customers arrive at a single-server station in accordance with a Poisson process with rate 4 per hour. The service times are exponentially distributed with rate 20 per hour. If each
40. Consider a M/G/1 system with λE[S] < 1.(a) Suppose that service is about to begin at a moment when there are n customers in the system.(i) Argue that the additional time until there are only n
39. Consider an M/G/1 system in which the first customer in a busy period has the service distribution G1 and all others have distribution G2. Let C denote the number of customers in a busy period,
38. For the M/G/1 queue, let Xn denote the number in the system left behind by the nth departure.(a) If
37. In an M/G/1 queue,(a) what proportion of departures leave behind 0 work?(b) what is the average work in the system as seen by a departure?
36. Compare the M/G/1 system for first-come, first-served queue discipline with one of last-come, first-served (for instance, in which units for service are taken from the top of a stack). Would you
35. Customers arrive at a single-server station in accordance with a Poisson process having rate λ. Each customer has a value. The successive values of customers are independent and come from a
34. For open queueing networks(a) state and prove the equivalent of the arrival theorem;(b) derive an expression for the average amount of time a customer spends waiting in queues.
33. Explain how a Markov chain Monte Carlo simulation using the Gibbs sampler can be utilized to estimate(a) the distribution of the amount of time spent at server j on a visit.Hint: Use the arrival
32. Consider a closed queueing network consisting of two customers moving among two servers, and suppose that after each service completion the customer is equally likely to go to either

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