All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Hire a Tutor
AI Study Help New

Search
Sign In
Register

study help
business
probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

22. Prove the following generalization of Exercise 21: n m m Cov (a; Xi, b;Y;) = a;b, Cov(X;, Y;). - i=1 j=1 i=1 j=1
21. Show that for random variables X, Y, Z, andW and constantsa, b,c, and d, Cov(aX + bY, cZ + dW)= ac Cov(X,Z) + bc Cov(Y,Z) + ad Cov(X,W) + bd Cov(Y,W).Hint: For a simpler proof, use the results of
20. Let S be the sample space of an experiment. Let A and B be two events of S. Let IA and IB be the indicator variables for A and B. That is,Show that IA and IB are positively correlated if and only
19. Let X be a random variable. Prove that Var(X) = mint E(X − t)2.Hint: Let μ = E(X) and look at the expansion of E(X − t)2= E(X − μ + μ − t)2.
18. Let X and Y be jointly distributed with joint probability density functionDetermine if X and Y are positively correlated, negatively correlated, or uncorrelated.Hint: Note that for all a > 0,
17. Find the variance of a sum of n randomly and independently selected points from the interval (0, 1).
16. Let X and Y have the following joint probability density function(a) Calculate Var(X + Y ).(b) Show that X and Y are not independent. Explain why this does not contradict Exercise 27 of Section
15. A voltmeter is used to measure the voltage of voltage sources, such as batteries. Every time this device is used, a random error is made, independent of other measurements, with mean 0 and
14. Let X and Y be independent random variables with expected values μ1 and μ2, and variances σ2 1 and σ2 2, respectively. Show that Var(XY)=0+
13. Mr. Jones has two jobs. Next year, he will get a salary raise of X thousand dollars from one employer and a salary raise of Y thousand dollars from his second. Suppose that X and Y are
8. For random variables X and Y, show that Cov(X + Y,X − Y ) = Var(X) − Var(Y ).9. Prove that Var(X − Y ) = Var(X) + Var(Y ) − 2 Cov(X, Y ).10. Let X and Y be two independent random
5. In n independent Bernoulli trials, each with probability of success p, let X be the number of successes and Y the number of failures. Calculate E(XY ) and Cov(X, Y ).
4. Thieves stole four animals at random from a farm that had seven sheep, eight goats, and five burros. Calculate the covariance of the number of sheep and goats stolen.
3. Roll a balanced die and let the outcome be X. Then toss a fair coin X times and let Y denote the number of tails. Find Cov(X, Y ) and interpret the result.Hint: Let p(x, y) be the joint
2. Let the joint probability mass function of random variables X and Y be given byFind Cov(X, Y ). p(x,y) = 0 x(x+y) if x = 1,2,3, y=3,4 elsewhere.
1. Ann cuts an ordinary deck of 52 cards and displays the exposed card. After Ann places her stack back on the deck, Andy cuts the same deck and displays the exposed card.Counting jack, queen, and
Using relation (10.10), calculate the variance of a negative binomial random variable X, with parameter (r, p).
Let X be the lifetime of an electronic system and Y be the lifetime of one of its components. Suppose that the electronic system fails if the component does (but not necessarily vice versa).
Ann cuts an ordinary deck of 52 cards and displays the exposed card. Andy cuts the remaining stack of cards and displays his exposed card. Counting jack, queen, and king as 11, 12, and 13, let X and
There are 300 cards in a box numbered 1 through 300. Therefore, the number on each card has one, two, or three digits. A card is drawn at random from the box. Suppose that the number on the card has
2. Let X be the number of students in Dr. Brown-Rose’s English 101 who will fail the course next semester. Show thatHint: Letand apply the Cauchy–Schwarz inequality P(X = 0) < Var(X) E(X2)
1. Shiante does not remember in which one of her 11 disk storage wallets she stored the last DVD that she purchased. Determine the expected number of the wallets that she should search to find the
19. Under what condition does Cauchy–Schwarz’s inequality become equality?
18. From an urn that contains a large number of red and blue chips, mixed in equal proportions, 10 chips are removed one by one and at random. The chips that are removed before the first red chip are
17. Let {X1,X2, . . .} be a sequence of continuous, independent, and identically distributed random variables. LetFind E(N). N = min{n: X> X2 X3 >> Xn-1, Xn-1
16. Let X and Y be nonnegative random variables with an arbitrary joint distribution function.Let(a) Show that(b) By calculating expected values of both sides of part (a), prove thatNote that this is
15. From an ordinary deck of 52 cards, cards are drawn at random, one by one, and without replacement until a heart is drawn. What is the expected value of the number of cards drawn?Hint: See
14. There are 25 students in a probability class. What is the expected number of the days of the year that are birthdays of at least two students? Assume that the birthrates are constant throughout
13. There are 25 students in a probability class. What is the expected number of birthdays that belong only to one student? Assume that the birthrates are constant throughout the year and that each
12. Suppose that 80 balls are placed into 40 boxes at random and independently.What is the expected number of the empty boxes?
11. A coin is tossed n times (n > 4). What is the expected number of exactly three consecutive heads?Hint: Let E1 be the event that the first three outcomes are heads and the fourth outcome is
10. Let {X1,X2, . . . ,Xn} be a set of independent random variables with P(Xj = i) = pi (1 ≤ j ≤ n and i ≥ 1). Let hk =P∞ i=k pi. Using Theorem 10.2, prove that E[min(X1, X2,..., Xn)] = h k=1
9. Solve the following problem posed by Michael Khoury, U.S. Mathematics Olympiad Member, in “The Problem Solving Competition,” Oklahoma Publishing Company and the American Society for
8. Let X1, X2, . . . , Xn be positive, identically distributed random variables. For 1 ≤ i ≤ n, letShow that Z1, Z2, . . . , Zn are also identically distributed and, for 1 ≤ i ≤ n, find
7. A cultural society is arranging a party for its members. The cost of a band to play music, the amount that the caterer will charge, the rent of a hall to give the party, and other expenses (in
6. An absentminded professor wrote n letters and sealed them in envelopes without writing the addresses on the envelopes. Having forgotten which letter he had put in which envelope, he wrote the n
5. A company puts five different types of prizes into their cereal boxes, one in each box and in equal proportions. If a customer decides to collect all five prizes, what is the expected number of
4. Let the joint probability density function of random variables X and Y beFind E(X), E(Y ), and E(X2 + Y 2). f(x, y): 2e-(x+2y) if x 0, y 0 === 0 otherwise.
3. Let X, Y, and Z be three independent random variables such that E(X) = E(Y ) =E(Z) = 0, and Var(X) =Var(Y ) =Var(Z) = 1. Calculate E[X2(Y + 5Z)2].
2. A calculator is able to generate random numbers from the interval (0, 1). We need five random numbers from (0, 2/5). Using this calculator, how many independent random numbers should we generate,
1. Let the probability density function of a random variable X be given byFind E(X2 + X). f(x)= | 1| if 0 x 2 otherwise.
A box contains nine light bulbs, of which two are defective. What is the expected value of the number of light bulbs that one will have to test (at random and without replacement) to find both
Dr. Windler’s secretary accidentally threw a patient’s file into the wastebasket.A few minutes later, the janitor cleaned the entire clinic, dumped the wastebasket containing the patient’s file
Exactly n married couples are living in a small town. What is the expected number of intact couples after m deaths occur among the couples? Assume that the deaths occur at random, there are no
A well-shuffled ordinary deck of 52 cards is divided randomly into four piles of 13 each. Counting jack, queen, and king as 11, 12, and 13, respectively, we say that a match occurs in a pile if the
A die is rolled 15 times. What is the expected value of the sum of the outcomes?
5. Suppose that 20% of the physicians working for a certain hospital retire before age 65, 30% retire at ages 65-69, and the remaining 50% retire at age 70 or later. If physicians retire
4. A system has 7 components, and it functions if and only if at least one of its components functions. Suppose that the lifetimes of the components are independent, identically distributed
3. For what value of c is the following a joint probability density function of four random variables X, Y, Z, and T? For that value of c find P(X x>0, y> 0, 2 > 0, t>0 (1+x+y+z+1) 6 f(x, y, z, t) =
2. Let X1 be a random point from the interval (0, 1), X2 be a random point from the interval (0,X1), X3 be a random point from the interval (0,X2), · · · , and Xn be a random point from the
1. Let (X, Y,Z) be a point selected at random in the unit sphere(x, y, z) : x2 + y2 + z2 ≤ 1;that is, the sphere of radius 1 centered at the origin. [Note that the volume of a sphere with radius R
10. Let X1, X2, and X3 be independent random variables from (0, 1). Find the probability density function and the expected value of the midrange of these random variables[X(1) + X(3)]/2.
8. A system consists of n components whose lifetimes form an independent sequence of random variables. Suppose that the system functions as long as at least one of its components functions. Let F1,
7. A system consists of n components whose lifetimes form an independent sequence of random variables. In order for the system to function, all components must function. Let F1, F2, . . . , Fn be the
6. Alvie, a marksman, fires seven independent shots at a target. Suppose that the probabilities that he hits the bull’s-eye, he hits the target but not the bull’s-eye, and he misses the target
4. The joint probability density function of random variables X, Y, and Z is given by(a) Determine the value of c.(b) Find P(X (c(x + y +22) if 0x, y, z 1 f(x, y, z) = 10 otherwise.
3. Suppose that n points are selected at random and independently inside the cube=(x, y, z) : − a ≤ x ≤a, −a ≤ y ≤a, −a ≤ z ≤ a.Find the probability that the distance of the
2. Let X be the smallest number obtained in rolling a balanced die n times. Calculate the distribution function and the probability mass function of X.
1. An urn contains 100 chips of which 20 are blue, 30 are red, and 50 are green. Suppose that 20 chips are drawn at random and without replacement. Let B, R, and G be the number of blue, red, and
2. Of the emails that arrive in Samantha’s inbox, 45% are personal, 40% are work-related, and 15% are unsolicited commercial emails. Samantha logs into her email account and finds that she has 25
1. Suppose that 40% of the students joining the mathematics department of a certain university major in actuarial science, 35% major in statistics and operation research, and 25% major in pure
8. Customers enter a department store at the rate of three per minute, in accordance with a Poisson process. If 30% of them buy nothing, 20% pay cash, 40% use charge cards, and 10% write personal
7. Suppose that the ages of 30% of the teachers of a country are over 50, 20% are between 40 and 50, and 50% are below 40. In a random committee of 10 teachers from this country, two are above
6. Suppose that 50% of the watermelons grown on a farm are classified as large, 30% as medium, and 20% as small. Joanna buys five watermelons at random from this farm.What is the probability that (a)
5. Of the drivers who are insured by a certain insurance company and get into at least one accident during a random year, 15% are low-risk drivers, 35% are moderate-risk drivers, and 50% are
4. At a certain college, 16% of the calculus students get A’s, 34% B’s, 34% C’s, 14% D’s, and 2% F’s. What is the probability that, of 15 calculus students selected at random, five get
3. Suppose that each day the price of a stock moves up 1/8 of a point with probability 1/4, remains the same with probability 1/3, and moves down 1/8 of a point with probability 5/12. If the price
2. An urn contains 100 chips of which 20 are blue, 30 are red, and 50 are green. We draw 20 chips at random and with replacement. Let B, R, and G be the number of blue, red, and green chips,
1. Light bulbs manufactured by a certain factory last a random time between 400 and 1200 hours.What is the probability that, of eight such bulbs, three burn out before 550 hours, two burn out after
Let X1, X2, . . . , Xr (r ≥ 4) have the joint multinomial probability mass function p(x1, x2, . . . , xr) with parameters n and p1, p2,. . . , pr. Find the marginal probability mass functions pX1
A warehouse contains 500 TV sets, of which 25 are defective, 300 are in working condition but used, and the rest are brand new.What is the probability that, in a random sample of five TV sets from
In a certain town, at 8:00 P.M., 30% of the TV viewing audience watch the news, 25% watch a certain comedy, and the rest watch other programs.What is the probability that, in a statistical survey of
2. All that we know about a horse race that was held last week in Louisville, Kentucky, is that all of the five horses that were competing reached the finish line, independently, at random times
1. Find the expected value of the distance between two random points selected independently from the interval (0, 1).
11. Let X1,X2, . . . ,Xn be n independently randomly selected points from the interval(0, θ), θ > 0. Prove thatwhere R = X(n) − X(1) is the range of these points.Hint: Use part (a) of Exercise
10. Let X1,X2, . . . ,Xn be a random sample of size n from a population with continuous distribution function F and probability density function f.(a) Calculate the probability density function of
9. Let X1 and X2 be two independent N(0, σ2) random variables. Find E[X(1)].Hint: Let f12(x, y) be the joint probability density function of X(1) and X(2). The desired quantity is RR xf12(x, y) dx
8. Let X1 and X2 be two independent exponential random variables each with parameterλ. Show that X(1) and X(2) − X(1) are independent.
7. Prove that G, the distribution function of [X(1) + X(n)]2, the midrange of a random sample of size n from a population with continuous distribution function F and probability density functionf, is
6. Let X1, X2, X3, . . . , Xm be a sequence of nonnegative, independent binomial random variables, each with parameters (n, p). Find the probability mass function of X(i), 1 ≤ i ≤ m.
5. Let X1, X2, X3, . . . , Xn be a sequence of nonnegative, identically distributed, and independent random variables. Let F be the distribution function of Xi, 1 ≤ i ≤ n.Prove thatHint: Use
4. Let X1, X2, X3, and X4 be independent exponential random variables, each with parameter λ. Find P(X(4) ≥ 3λ).
3. A box contains 20 identical balls numbered 1 to 20. Seven balls are drawn randomly and without replacement. Find the probability mass function of the median of the numbers on the balls drawn.
The distance between two towns, A and B, is 30 miles. If three gas stations are constructed independently at randomly selected locations between A and B, what is the probability that the distance
Suppose that a machine consists of n components with the lifetimes X1, X2,. . . , Xn, respectively, where Xi’s are independent and identically distributed. Suppose that the machine remains
Suppose that customers arrive at a warehouse from n different locations. Let Xi, 1 ≤ i ≤ n, be the time until the arrival of the next customer from location i; then X(1) is the arrival time of
2. Let X, Y, and Z be continuous random variables with the joint probability density function given byFind P(X 2y e-(2x+y+2) x>0, y>0, 2>0 f(x, y, z) = otherwise.
1. In the inventory of a pharmacy, in a carton, there are 40 boxes of painkillers of which 15 are brand A, 10 are brand B, 6 are brand C, 4 are brand D, and 5 are brand E. An assistant pharmacist
26. (Roots of Cubic Equations) Solve the following exercise posed by S. A. Patil and D. S. Hawkins, Tennessee Technological University, Cookeville, Tennessee, in The College Mathematics Journal,
25. (Roots of Quadratic Equations) Three numbers A, B, and C are selected at random and independently from the interval (0, 1). Determine the probability that the quadratic equation Ax2 + Bx + C = 0
24. A point is selected at random from the pyramid V =(x, y, z) : x, y, z ≥ 0, x + y + z ≤ 1.Letting (X, Y,Z) be its coordinates, determine if X, Y, and Z are independent.Hint: Recall that the
23. Suppose that h is the probability density function of a continuous random variable. Let the joint probability density function of X, Y, and Z be f(x, y, z) = h(x)h(y)h(z), x, y, z ∈ R.Prove
22. Let X1,X2, . . . ,Xn be n independent random numbers from (0, 1), and Yn = n · min(X1,X2, . . . ,Xn).Prove that lim n→∞P(Yn > x) = e−x, x ≥ 0.
21. Let X1,X2, . . . ,Xn be n independent random numbers from the interval (0, 1). Find E????max 1≤i≤n Xiand E????min 1≤i≤n Xi.
20. Let F be a distribution function. Prove that the functions Fn and 1 − (1 − F)n are also distribution functions.Hint: Let X1,X2, . . . ,Xn be independent random variables each with the
19. (Reliability of Systems) To transfer water from point A to point B, a water-supply system with five water pumps located at the points 1, 2, 3, 4, and 5 is designed as in Figure 9.5. Suppose that
18. (Reliability of Systems) Consider the system whose structure is shown in Figure 9.4. Find the reliability of this system. 1 2 3 4 5 6 7 Figure 9.4 A diagram for the system of Exercise 18.
17. Suppose that the lifetimes of a certain brand of transistor are identically distributed and independent random variables with distribution function F. These transistors are randomly selected, one
16. An item has n parts, each with an exponentially distributed lifetime with mean 1/λ. If the failure of one part makes the item fail, what is the average lifetime of the item?Hint: Use the result
15. (Reliability of Systems) Suppose that a system functions if and only if at least k (1 ≤ k ≤ n) of its components function. Furthermore, suppose that pi = p for 1 ≤ i ≤ n. Find the

Showing 300 - 400 of 6914

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Last

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Secure Payment

Download Our App

© 2024 SolutionInn. All Rights Reserved