Introduction To Probability Models 6th Edition Sheldon M. Ross - Solutions

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12. On Ω=(a,b),−∞< a < b
11. A lot of five identical batteries is life tested. The probability assignment is assumed to be P(A) =A(1/λ)e−x/λdx for any event A⊆ [0,∞), whereλ>0 is a known constant. Thus the probability that a battery fails after time t is given by P(t,∞) = ∞t(1/λ)e−x/λdx, t ≥ 0.If the
10. Show that the converse of Theorem 2 also holds. Thus A and B are independent if, and only if, A and Bc are independent, and so on.
9. Let (Ω,S,P) be a probability space. Let A, B, C ∈ S with PB and PC > 0. If B and C are independent show that P{A | B} = P{A | B∩C}PC+P{A | B∩Cc}PCc.Conversely, if this relation holds, P{A | BC} = P{A | B}, and PA > 0, then B and C are independent (Strait [111]).
8. A pond contains red and golden fish. There are 3000 red and 7000 golden fish, of which 200 and 500, respectively, are tagged. Find the probability that a random sample of 100 red and 200 golden fish will show 15 and 20 tagged fish, respectively.
7. Let A, B, and C be three boxes with three, four, and five cells, respectively. There are three yellow balls numbered 1 to 3, four green balls numbered 1 to 4, and five red balls numbered 1 to 5. The yellow balls are placed at random in box A, the green in B, and the red in C, with no cell
6. An urn contains four red and four black balls. A sample of two balls is drawn at random. If both balls drawn are of the same color, these balls are set aside and a new sample is drawn. If the two balls drawn are of different colors, they are returned to the urn and another sample is drawn.
5. A box contains 20 black and 30 green balls. One ball at a time is drawn at random, its color is noted, and the ball is then replaced in the box for the next draw.(a) Find the probability that the first green ball is drawn on the fourth draw.(b) Find the probability that the third and fourth
4. A biased coin with probability p, 0 < p < 1, of success (heads) is tossed until for the first time the same result occurs three times in succession (i.e., three heads or three tails in succession). Find the probability that the game will end at the seventh throw.
3. Let A1, A2, and A3 be three independent events. Show that Ac 1, Ac 2, and Ac 3 are independent.
2. Let A and B be two independent events defined on some probability space, and let PA = 1/3, PB = 3/4. Find (a) P(A∪B), (b) P{A | A∪B}, and (c) P{B | A∪B}.
3. Let X be a positive RV of the continuous type with PDF f (·). Find the PDF of the RV U = X/(1+X). If, in particular, X has the PDF f (x) =1, 0 ≤ x ≤ 1, 0, otherwise, what is the PDF of U?
5. Let X be an RV with PDF fθ(x) =θe−θx if x ≥ 0, 0 otherwise, where θ > 0. Let Y = [X−1/θ]2. Find the PDF of Y.
6. A point is chosen at random on the circumference of a circle of radius r with center at the origin, that is, the polar angle θ of the point chosen has the PDF f (θ) =1 2π, θ∈ (−π,π).Find the PDF of the abscissa of the point selected.
8. Let X be an RV such that P(a ≤ X ≤b) = 1 where −∞ < a < b < ∞. Show that var(X) ≤ (b−a)2/4.
7. For any RV X, show that P{X ≥ 0} ≤ inf{ϕ(t) : t ≥ 0} ≤ 1, where ϕ(t) = EetX , 0
6. Use Chebychev’s inequality to show that for any k > 1, ek+1 ≥ k2.
5. Let X be an RV with EX = 0, var(X) = σ2, and EX4 = μ4. Let K be any positive real number. Show that P{|X| ≥ Kσ} ≤⎧⎪⎪⎨⎪⎪⎩1 if K2 < 1, 1 K2 if 1 ≤ K2 < μ4σ4 ,μ4−σ4μ4+σ4K4−2K2σ4 if K2 ≥ μ4σ4 .In other words, show that bound (7) is better than bound (3) if K2
4. Let g(.) be a function satisfying g(x) > 0 for x > 0, g(x) increasing for x > 0, and E|g(X)| ε} 0.
3. Construct an example to show that inequalities (4) and (5) cannot be improved.
2. Let X be any RV, and suppose that the MGF of X,M(t) = Eetx, exists for every t >0.Then for any t > 0 P{tX > s2 +logM(t)} < e−s2.
1. For the RV with PDF f (x;λ) =e−xxλλ!, x > 0, where λ ≥ 0 is an integer, show that P{0 < X < 2(λ+1)} >λλ+1.
12. Let X be an RV of the continuous type, and let Y = g(X) be defined as follows:(a) g(x) = 1 if x > 0, and = −1 if x ≤ 0.(b) g(x) = b if x ≥b, = x if |x|
11. Let X be an RV with PDF f (x) = 1/(2θ) if −θ ≤ x ≤ θ, and = 0 otherwise. Let Y = 1/X2. Find the PDF of Y.
10. Let X be an RV with PDF f (x)=1/3 if−1
9. Let X be an RV with PDF f (x) = 1/(2π) if 0 < x < 2π, and = 0 otherwise. Let Y = sinX. Find the DF and PDF of Y.
8. Suppose that a projectile is fired at an angle θ above the earth with a velocity V.Assuming that θ is an RV with PDF f (θ) =⎧⎨⎩12πifπ6< θ
7. For the RV X of Example 7 find the PDF of the following RVs: (a) Y1 =eX, (b) Y2 =2X2 +1, and (c) Y3 = g(X), where g(x) = 1 if x > 0, = 1/2 if x = 0, and = −1 if x < 0.
1. A biased coin is tossed until a head appears for the first time. Let p be the probability of a head, 0 < p < 1. What is the probability that the number of tosses required is odd? Even?
9. Let X1,X2 be iid with PDF fσ(x) =2σ1− xσ, 0 < x
4. Let X be an RV with PDF f defined by Example 11. Let Y = cosX and Z = tanX.Find the DFs and PDFs of Y and Z.
5. In an experiment to determine whether four different makes of automobiles average the same gasoline mileage, a random sample of two cars of each make was taken from each of four cities. Each car was then test run on 5 gallons of gasoline of the same brand. The following table gives the number of
1. Let (X1,X2, . . . ,Xn) be jointly normal with EXi = μ, var(Xi) = σ2, and cov(Xi,Xj) =ρσ2 if |i−j| = 1, i = j, and = 0 otherwise.(a) Show that var(X) =σ2 n1+2ρ1− 1 nand E(S2) = σ21− 2ρn.(b) Show that the t-statistic√n(X−μ)/S is asymptotically normally distributed with mean
2. Let X1,X2, . . . ,Xn be a random sample from G(α,β), α>0, β >0:(a) Show thatμ4 = 3α(α+2)/β4.(b) Show that var(n−1)S2σ2≈ (n−1)2+6α.(c) Show that the large sample distribution of (n−1)S2/σ2 is normal.(d) Compare the large-sample test of H0 : σ = σ0 based on the asymptotic
3. Let X1,X2, . . . ,Xm and Y1,Y2, . . . ,Yn be two independent random samples from populations with means μ1 and μ2, and variances σ2
1 and σ2
2, respectively. Let X,Y be the two sample means, and S2 1,S2 2 be the two sample variances. Write N = m+n, R = m/n, and θ = σ2 1/σ2
2. The usual normal theory test of H0 : μ1 −μ2 = δ0 is the t-test based on the statistic T =X−Y −δ0 Sp(1/m+1/n)1/2 , where S2 p =(m−1)S2 1 +(n−1)S2 2m+n−2.Under H0, the statistic T has a t-distribution with N −2 d.f., provided that σ2 1 = σ2
2.Show that the asymptotic distribution of T in the nonnormal case is N(0, (θ+R)(1 + Rθ)−1) for large m and n. Thus, if R = 1, T is asymptotically N(0,1) as in the normal theory case assuming equal variances, even though the two samples come from nonnormal populations with unequal variances.
4. Verify the computations in the table above using the expressions of ARE in (28),(29), and (30).
5. Suppose F is a G(α,β) r.v. Show that e(W, X) =3αΓ2(2α)24(α−1)(2α−1)2{Γ(α)}4 .(Note that F is not symmetric.)
4. Suppose that the following data represent the units of production turned out each day by three different machinists, each working on the same machine for three different days:Machinist Machine A B C B1 15, 15, 17 19, 19, 16 16, 18, 21 B2 17, 17, 17 15, 15, 15 19, 22, 22 B3 15, 17, 16 18, 17, 16
3. Prove statement (10).
2. Obtain the likelihood ratio test of the null hypothesis Hβ : β1 = β2 = · · · = βb = 0.
10. Let X1,X2, . . . ,Xn be iid with PDF fθ(x) =1θexp(−x/θ), x > 0, and = 0, otherwise.Find the Pitman estimator of θk.
1. Let X1,X2, . . . ,Xn be iid C(1,θ) RVs. Show that E0(1 + X2 1)−k = (1/π)B(k +1/2,1/2). Hence or otherwise show that E0!X2 1(1+X2 1 )2"= var#X1 1+X2 1$= 1/8.
2. Let X1,X2, . . . ,Xn be a random sample from logistic PDF fθ(x) =1 2[1+cosh(x−θ)]=ex−θ{1+ex−θ}2 .Show that the LMP test of H0 :θ=0 against H1 :θ>0 rejects H0 ifn% i=1 tanh xi 2&> k.
3. Let X1,X2, . . . ,Xn be iid RVs with common Laplace PDF fθ(x) = (1/2)exp{−|x−θ|}.For n ≥ 2 show that UMP size α (0< α 0 does not exist. Find the form of the LMP test.
1. In Example 1 let n = 15, μ0 = 4.7, and μ1 = 5.2, and choose a = b > 0. Find the minimax test, and compute its power at μ = 4.7 and μ = 5.2.
2. A sample of five observations is taken on a b(1,θ) RV to test H0 : θ = 12 against H1 : θ = 34.(a) Find the most powerful test of size α = 0.05.(b) If L( 1 2 , 12) = L( 3 4 , 34) = 0, L( 1 2 , 34) = 1, and L( 3 4 , 12) = 2, find the minimax rule.(c) If the prior probabilities of θ = 12 and
3. A sample of size n is to be used from the PDF fθ(x) = θe−θx, x > 0, to test H0 : θ =1 against H1 : θ =2. If the a priori distribution on θ is π0 = 23, π1 = 13, and a =b, find the Bayes solution. Find the power of the test at θ = 1 and θ = 2.
1. Prove statement (6).
6. For the multiple decision problem described in Remark 2 show that a Bayes solution is to accept Hi : θ = θi (i = 1,2, . . . .k) if (15) holds.
5. Given three normal densities with variances 1 and with means −1, 0, 1, respectively, find the Bayes solution to the multiple decision problem based on a single observation when π1 = 25, π2 = 25, π3 = 15.
4. Given two normal densities with variances 1 and with means −1 and 1, respectively, find the Bayes solution based on a single observation when a = b and (a) π0 =π1 = 12, and (b) π0 = 14, π1 = 34.
Consider a post office with two clerks. Three people, A, B, and C, enter simultaneously. A and B go directly to the clerks, and C waits until either A or B leaves before he begins service. What is the probability that A is still in the post office after the other two have left when (a) the service
The lifetime of a radio is exponentially distributed with a mean of ten years. If Jones buys a ten-year-old radio, what is the probability that it will be working after an additional ten years?
In Example 5.2 if server i serves at an exponential rate , i = 1, 2, show that P{Smith is not last)-(+) = 22 +
Show that the failure rate function of a gamma distribution with parameters n and A is increasing when n 1.
Norb and Nat enter a barbershop simultaneously-Norb to get a shave and Nat a haircut. If the amount of time it takes to receive a haircut (shave) is exponentially distributed with mean 20 (15) minutes, and if Norb and Nat are immediately served, what is the probability that Nat finishes before Norb?
If X and Y are independent exponential random variables with respective means 1/2, and 1/2, then (a) use the lack of memory property of the exponential to intuitively explain why Z min(X, Y) is exponential. (b) what is the conditional distribution of Z given that Z = X? (c) give a heuristic
In a certain system, a customer must first be served by server 1 and then by server 2.The service times at server i are exponential with rate , i=1,2. An arrival finding server 1 busy waits in line for that server. Upon completion of service at server 1, a customer either enters service with server
Suppose in Exercise 15 you arrive to find two others in the system, one being served by server 1 and one by server 2.What is the expected time you spend in the system? Recall that if server 1 finishes before server 2, then server 1's customer will remain with him (thus blocking your entrance) until
A flashlight needs two batteries to be operational. Consider such a flashlight along with a set of n functional batteries-battery 1, battery 2, ..., battery n. Initially, battery 1 and 2 are installed. Whenever a battery fails, it is immediately replaced by the lowest numbered functional battery
Argue that if ,, i=1,..., n are distinct positive numbers then 2j - = 1 Hint: Relate this problem to Section 5.2.4.
Show that Definition 5.1 of a Poisson process implies Definition 5.3.
Show that assumption (iv) of Definition 5.3 follows from assump- tions (ii) and (iii). Hint: Derive a functional equation for g(t) = P{N(t) = 0).
Cars cross a certain point in the highway in accordance with a Poisson process with rate = 3 per minute. If Reb blindly runs across the highway, then what is the probability that she will be uninjured if the amount of time that it takes her to cross the road is s seconds? (Assume that if she is on
Suppose in Exercise 29 that Reb is agile enough to escape from a single car, but if she encounters two or more cars while attempting to cross the road, then she will be injured. What is the probability that she will be unhurt if it takes her s seconds to cross. Do it for s = 5, 10, 20, 30.
Show that if (N,(t), t 0) are independent Poisson processes with rate 2, 1, 2, then [N(t), t 0} is a Poisson process with rate + A where N(t) = N(t) + N(1).
In Exercise 31 what is the probability that the first event of the combined process is from the N, process?
Let [N(), 0] be a Poisson process with rate 1.Let S,, denote the time of the nth event. Find (a) E[S4] (b) E[S, N(1) (c) E[N(4) 2] N(2) N(1) = 3]
Events occur according to a Poisson process with rate = 2 per hour. (a) What is the probability that no event occurs between 8 P.M. and 9 P.M.? (b) Starting at noon, what is the expected time at which the fourth event occurs? (c) What is the probability that two or more events occur between 6 P.M.
Pulses arrive at a Geiger counter in accordance with a Poisson process at a rate of three arrivals per minute. Each particle arriving at the counter has a probability of being recorded. Let X(r) denote the number of pulses recorded by time minutes. (a) P(X(r) = 0) = ? (b) E[X(t)] = ?
Cars pass a point on the highway at a Poisson rate of one per minute. If 5 percent of the cars on the road are vans, then (a) what is the probability that at least one van passes by during an hour? (b) given that ten vans have passed by in an hour, what is the expected number of cars to have passed
Customers arrive at a bank at a Poisson rate A. Suppose two customers arrived during the first hour. What is the probability that (a) both arrived during the first 20 minutes? (b) at least one arrived during the first 20 minutes?
A system has a random number of flaws that we will suppose is Poisson distributed with meanc. Each of these flaws will, independently, cause the system to fail at a random time having distribution G. When a system failure occurs, suppose that the flaw causing the failure is immediately located and
A cable car starts off with n riders. The times between successive stops of the car are independent exponential random variables with rate A. At each stop one rider gets off. This takes no time, and no additional riders get on. After a rider gets off the car, he or she walks home. Independently of
Shocks occur according to a Poisson process with rate , and each shock independently causes a certain system to fail with probability p. Let T denote the time at which the system fails and let N denote the number of shocks that it takes. (a) Find the conditional distribution of T given that N-n.
The number of missing items in a certain location, call it X, is a Poisson random variable with mean 2.When searching the location, each item will independently be found after an exponentially distributed time with rate u. A reward of R is received for each item found, and a searching cost of C per
Suppose that the times between successive arrivals of customers at a single-server station are independent random variables having a common distribution F. Suppose that when a customer arrives, he or she either immediately enters service if the server is free or else joins the end of the waiting
(a) Let [N(t), 1 0] be a nonhomogeneous Poisson process with mean value function m(r). Given N(t) =n, show that the unordered set of arrival times has the same distribution as n independent and identically distributed random variables having distribution function m(x) F(x)=m(t) x>1 (b) Suppose that
Suppose that events occur according to a nonhomogeneous Poisson process with intensity function (t), t 0.Suppose that, independently of anything that has previously occurred, an event at times will be counted with probability p(s), s 0.Let N, (f) denote the number of counted events by time t. (a)
Suppose that (No(), 0) is a Poisson process with rate = 1.Let (1) denote a nonnegative function of t, and let Define N(t) by m(t)=2(s) ds N(t) = No(m(t)) Argue that [N(t), t 0) is a nonhomogeneous Poisson process with intensity function A(1), 0.Hint: Make use of the identity m(t + h) m(t) = m'(t)h
Let X1, X2,... be independent and identically distributed non- negative continuous random variables having density function f(x). We say that a record occurs at time n if X, is larger than each of the previous values XX-1 (A record automatically occurs at time 1.) If a record occurs at time n, then
In Exercise 67 show that X and X2 both have exponential distributions.
Let X1, X2,..., X, be independent and identically distributed exponential random variables. Show that the probability that the largest of them is greater than the sum of the others is n/2"-1. That is if then show M = max X P{M> X-M} - 1-1 Hint: What is PIX> [2X]?
A two-dimensional Poisson process is a process of randomly occurring events in the plane such that (i) for any region of area A the number of events in that region has a Poisson distribution with mean AA and (ii) the number of events in nonoverlapping regions are independent.
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 probability Ah + o(h). Each mating immediately produces one offspring, equally likely to be male or female.
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 a; an individual in state B divides into two new individuals of type A at an exponential rate . Define an appropriate continuous-time Markov chain for
Consider two machines that are maintained by a single repairman. Machine functions for an exponential time with rate , before breaking down, i = 1,2. The repair times (for either machine) are exponential with rate . Can we analyze this as a birth and death process? If so, what are the parameters?
Potential customers arrive at a single-server station in accordance with a Poisson process with rate A. However, if the arrival finds n customers already in the station, then he will enter the system with probability a,.. Assuming an exponential service rate , set this up as a birth and death
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 having rate . When a contact occurs, it is equally likely to involve any of the pairs of individuals in
Consider a birth and death process with birth rates = (i + 1)2, i0, and death rates ;= iu, i 0.(a) Determine the expected time to go from state 0 to state 4.(b) Determine the expected time to go from state 2 to state 5.(c) Determine the variances in parts (a) and (b).
Individuals join a club in accordance with a Poisson process with rate A. Each new member must pass through k consecutive stages to become a full member of the club. The time it takes to pass through each stage is exponentially distributed with rate u. Let N,(t) denote the number of club members at
Consider two machines, both of which have an exponential lifetime with mean 1/2. There is a single repairman that can service machines at an exponential rate . Set up the Kolmogorov backward equations; you need not solve them.
The birth and death process with parameters ,, = 0 and = , n > 0 is called a pure death process. Find Pi(t).
Consider two machines. Machine i operates for an exponential time with rate , and then fails; its repair time is exponential with rate ,, i = 1, 2.The machines act independently of each other. Define a four-state continuous-time Markov chain which jointly describes the condition of the two
Consider a Yule process starting with a single individual-that is, suppose X(0) 1.Let 7, denote the time it takes the process to go from a population of size i to one of size i + 1.(a) Argue that T, =1,..., are independent exponentials with respective rates i. (b) Let X., X, denote independent
Each individual in a biological population is assumed to give birth at an exponential rate , and to die at an exponential rate u. In addition, there is an exponential rate of increase due to immigration. However, immi- gration is not allowed when the population size is N or larger. (a) Set this up

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Introduction To Probability Models 6th Edition Sheldon M. Ross - Solutions

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