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

1. Is the following the probability density function of some beta random variableX? If so, find E(X) and Var(X). 12x(1-x)2 0
Suppose that all 25 passengers of a commuter plane, departing at 2:00 P.M., arrive at random times between 12:45 and 1:45 P.M. Find the probability that the median of the arrival times of the
The proportion of the stocks that will increase in value tomorrow is a beta random variable with parameters α and β. Today, these parameters were determined to beα = 5 and β = 4, by a financial
2. A company is behind in manufacturing upper fuser rolling bearings for its photocopiers and needs to produce 500 of them right away. If the company produces these items at a Poisson rate of 5 per
1. A vacuum tube is a device that facilitates the free passage of electric current between electrodes in an evacuated container. The lifetime, in thousands of hours, of a vacuum tube manufactured by
14. In data communication,messages are usually combinations of characters, and each character consists of a number of bits. A bit is the smallest unit of information and is either 1 or 0. Suppose
13. Howard enters a bank that has n tellers. All the tellers are busy serving customers, and there is exactly one queue being served by all tellers, with one customer ahead of Howard waiting to be
12. (a) Let Z be a standard normal random variable. Show that the random variable Y =Z2 is gamma and find its parameters.(b) Let X be a normal random variable with mean μ and standard deviation σ.
11. For n = 0, 1, 2, 3, . . . , calculate ????(n + 1/2).
10. The lifetime of the alarm system installed in Herman Hall at a private university is exponential with mean 1/λ. Beginning at time T, the facility management crew inspects the system every T
9. A manufacturer produces light bulbs at a Poisson rate of 200 per hour. The probability that a light bulb is defective is 0.015. During production, the light bulbs are tested one by one, and the
8. Suppose that the lifetime of a certain wheel bearing manufactured by a company is a gamma random variable with mean 75,000 and standard deviation 15,000 miles. If the manufacturer offers a free
7. Experience has shown that the bugs of a certain smartphone app cause battery drainage.The time that it will take for the app developers to locate and remove the bugs is a gamma randomvariable with
6. Customers arrive at a restaurant at a Poisson rate of 12 per hour. If the restaurant makes a profit only after 30 customers have arrived, what is the expected length of time until the restaurant
5. Let f be the probability density function of a gamma random variable X, with parameters(r, λ). Prove that R∞−∞f(x) dx = 1.
There are 100 questions in a test. Suppose that, for all s > 0 and t > 0, the event that it takes t minutes to answer one question is independent of the event that it takes s minutes to answer
Suppose that, on average, the number of β-particles emitted from a radioactive substance is four every second.What is the probability that it takes at least 2 seconds before the next two
2. Patients arrive at a pharmacy for flu shots at a Poisson rate of 3 per hour. If the last patient, who had a flu shot at the pharmacy, arrived 45 minutes ago, what is the probability that the next
1. When solid state hard drives fail to function, it is not possible to repair them. They should be replaced. Such drives, manufactured by Seaportal Technology, have lifetimes that are exponentially
18. Prove that if X is a positive, continuous, memoryless random variable with distribution function F, then F(t) = 1 − e−λt for some λ > 0. This shows that the exponential is the only
17. Let X, the lifetime (in years) of a radio tube, be exponentially distributed with mean 1/λ. Prove that [X], the integer part of X, which is the complete number of years that the tube works, is a
16. The random variable X is said to be a Laplace random variable or double exponentially distributed if its probability density function is given by(a) Find the value of c.(b) Prove that E(X2n) =
15. In data communication,messages are usually combinations of characters, and each character consists of a number of bits. A bit is the smallest unit of information and is either 1 or 0. Suppose
14. In a factory, a certain machine operates for a period which is exponentially distributed with parameter λ. Then it breaks down and will be in the repair shop for a period, which is also
13. The service times at a two-teller bank are independent exponential random variables each with parameter λ. At a random time Jim is being served by teller 2, Jeff is being served by teller 1,
12. Mr. Jones is waiting to make a phone call at a train station. There are two public telephone booths next to each other, occupied by two persons, say A and B. If the duration of each telephone
11. The profit is $350 for each computer assembled by a certain person. Suppose that the assembler guarantees his computers for one year and the time between two failures of a computer is exponential
10. The time between consecutive arrival times of two buses at a station is exponential with mean 20 minutes. Every day, during the operating hours of the buses, Kayla arrives at the station at a
9. Suppose that the time it takes for a novice secretary to type a document is exponential with mean 1 hour. If at the beginning of a certain eight-hour working day the secretary receives 12
8. Suppose that, at an Italian restaurant, the time, in minutes, between two customers ordering pizza is exponential with parameter λ. What is the probability that (a) no customer orders pizza
7. Suppose that losses under an insurance policy are distributed exponentially with mean$6000. If the amount of deductible for each loss is $500, what is the median of the insurance company’s claim
6. Let X be an exponential random variable with parameter λ. Find P(X E(X) 20x).
5. Guests arrive at a hotel, in accordance with a Poisson process, at a rate of five per hour.Suppose that for the last 10 minutes no guest has arrived. What is the probability that(a) the next one
Suppose that, on average, two earthquakes occur in San Francisco and two in Los Angeles every year. If the last earthquake in San Francisco occurred 10 months ago and the last earthquake in Los
The lifetime of a TV tube (in years) is an exponential random variable with mean 10. If Jim bought his TV set 10 years ago, what is the probability that its tube will last another 10 years?
At an intersection there are two accidents per day, on average. What is the probability that after the next accident there will be no accidents at all for the next two days?
Suppose that every three months, on average, an earthquake occurs in California. What is the probability that the next earthquake occurs after three but before seven months?
2. Every day a factory produces 5000 light bulbs, of which 2500 are type I and 2500 are type II. If a sample of 40 light bulbs is selected at random to be examined for defects, what is the
1. The grades of students in a calculus-based probability course are normal with mean 72 and standard deviation 7. If 90, 80, 70, and 60 are the respective lowest, A, B, C, and D, what percent of
34. Let I =R∞0 e−x2/2 dx; thenLet y/x = s and change the order of integration to show that I2 = π/2. This gives an alternative proof of the fact that is a distribution function. The advantage
33. In a forest, the number of trees that grow in a region of area R has a Poisson distribution with mean λR, where λ is a positive real number. Find the expected value of the distance from a
32. At an archaeological site 130 skeletons are found and their heights are measured and found to be approximately normal with mean 172 centimeters and variance 81 centimeters.At a nearby site, five
31. The amount of soft drink in a bottle is a normal random variable. Suppose that in 7%of the bottles containing this soft drink there are fewer than 15.5 ounces, and in 10% of them there are more
30. Let Z be a standard normal random variable. Show that for x > 0,Hint: Use part (b) of Exercise 29. I lim P(Z >t+ | Zt) = e. - t
29. (a) Prove that for all x > 0,Hint: Integrate the following inequalities(b) Use part (a) to prove that 1 − (x) ∼1 x√2πe−x2/2. That is, as x → ∞, the ratio of the two sides
28. Prove that for some constant k, f(x) = ka−x2, a ∈ (0,∞), is a normal probability density function.
27. To examine the accuracy of an algorithm that selects random numbers from the set{1, 2, . . . , 40}, 100,000 numbers are selected and there are 3500 ones. Given that the expected number of ones is
26. Suppose that the odds are 1 to 5000 in favor of a customer of a particular bookstore buying a certain fiction bestseller. If 800 customers enter the store every day, how many copies of that
25. Let X ∼ N(0, 1). Calculate the probability density function of Y =p|X|.
24. Let X ∼ N(μ, σ2). Calculate the probability density function of Y = eX.
23. Let X ∼ N(0, σ2). Calculate the probability density function of Y = X2.
22. Let α ∈ (−∞,∞) and Z ∼ N(0, 1); find E(eαZ).
19. Let X ∼ N(μ, σ2). Find the distribution function of |X − μ| and its expected value.
18. Find the expected value and the variance of a random variable with the probability density function f(x) = 2 -e -2(x-1) -x < < .
17. The annual rate of return for a share of a specific stock is a normal random variable with mean 0.12 and standard deviation of 0.06. The current price of the stock is $35 per share. Mrs. Lovotti
5. Let X be a standard normal random variable. Calculate E(X cosX), E(sinX), and E X 1+X2
4. Let (x) = 2(x) − 1. The function is called the positive normal distribution.Prove that if Z is standard normal, then |Z| is positive normal.
3. A small college has 1095 students. What is the approximate probability that more than five students were born on Christmas day? Assume that the birthrates are constant throughout the year and that
2. Suppose that 90% of the patients with a certain disease can be cured with a certain drug.What is the approximate probability that, of 50 such patients, at least 45 can be cured with the drug?
1. Let X ∼ N(−2, 5). Find P????|X| < 4.
The annual rate of return for a share of a specific stock is a normal random variable with mean 10% and standard deviation 12%. Ms. Couture buys 100 shares of the stock at a price of $60 per share.
The scores on an achievement test given to 100,000 students are normally distributed with mean 500 and standard deviation 100. What should the score of a student be to place him among the top 10% of
Suppose that of all the clouds that are seeded with silver iodide, 58% show splendid growth. If 60 clouds are seeded with silver iodide, what is the probability that exactly 35 show splendid growth?
2. Suppose that X, the error of a measurement, is a random number between −1 and 1.Find the expected value and the variance of |X|, the magnitude of the error.Hint: By calculating the distribution
1. A point is selected at random on a line segment of length ℓ. What is the probability that none of the two segments is smaller than ℓ/3?
16. RThe sample space of an experiment is S = (0, 1), and for every subset A of S, P(A) =A dx. Let X be a random variable defined on S by X(ω) = 5ω −1. Prove that X is a uniform random variable
15. Let g be a nonnegative real-valued function on R that R satisfies the relation∞−∞g(t) dt = 1. Show that if, for a random variable X, the random variable Y =R X−∞g(t) dt is uniform, then
13. Let X be a uniform random variable over the interval (0, 1 + θ), where 0 < θ < 1 is a given parameter. Find a function of X, say g(X), so that Eg(X)= θ2.
12. Let X be a random number from (0, 1). Find the probability density functions of the random variables Y = −ln(1 − X) and Z = Xn, n 6= 0.
11. Let X be a random number from [0, 1]. Find the probability mass function of [nX], the greatest integer less than or equal to nX.
9. A farmer who has two pieces of lumber of lengths a and b (a
7. The loss for an insurance policy, in thousands of dollars, denoted by X, is a random amount between 0 and 10. The insurance company wants to determine the amount of deductible under this policy so
1.What is the probability that a random chord of a circle is longer than a side of an equilateral triangle inscribed into the circle?
A person arrives at a bus station every day at 7:00 A.M. If a bus arrives at a random time between 7:00 A.M. and 7:30 A.M., what is the average time spent waiting?
2. Using Monte Carlo procedure, write a program to estimate -(3) − -(−1) = E 3 −1 1 √2π e−x2/2 dx. Then run your program for n = 10, 000 and compare the answer with that obtained from
7. Mr. Jones is at a train station, waiting to make a phone call. There are two public telephone booths next to each other and occupied by two persons, say A and B. If the duration of each telephone
6. Suppose that in a community the distributions of the heights of men and women, in centimeters, are N (173, 40) and N (160, 20), respectively. Write an algorithm to calculate by simulation the
5. It can be shown that the median of (2n + 1) random numbers from the interval (0, 1)is a beta random variable with parameters(n+1, n+1). Use this property to simulate a beta random variable with
4. Use the result of Example 11.11 to explain how a gamma random variable with parameters (n/2, 1/2), n being a positive integer, can be simulated.
3. Explain a procedure for simulation of lognormal random variables. A random variable X is called lognormal with parameters µ and σ2 if ln X ∼ N (µ, σ2)
2. Explain how a random variable X with the following probability density function can be simulated: f (x) = e−2|x| , −∞ < x < +∞.
1. Let X be a random variable with probability distribution function F (x) =    x − 3 x − 2 if x ≥ 3 0 elsewhere. Develop a method to simulate X.
4. TargetsA and B are placed on a wall. It is known that for every shot the probabilities of hitting A and hitting B with a missile are, respectively, 0.3 and 0.4. If target A was not hit in an
3. In Example 13.5, Laplace’s law of succession, suppose that, among the first m balls drawn, r are red and m − r are white. Using computer simulation, find the approximate probability that the
2. From families with five children, a family is selected at random and found to have a boy. Using computer simulation, find the approximate value of the probability that the family has three boys
1. An ordinary deck of 52 cards is dealt among A, B, C, and D, 13 each. If A and B have a total of six hearts and five spades, using computer simulation find an approximate value for the probability
22. Let V (t) be the price of a stock, per share, at time t. Suppose that $ V (t): t ≥ 0 % is a geometric Brownian motion with drift parameter $3 per year and variance parameter 27.04. What is the
21. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. For t > 0, let T be the smallest zero greater than t, and let U be the largest zero smaller than t. For x < t < y, find P
20. Let $ X(t): t ≥ 0 % be a Brownian motion with variance parameter σ2. Fort, s > 0, find E 4 X(s)X(t)5 .
19. Suppose that liquid in a cubic container is placed in a coordinate system, and at time 0, a pollen particle suspended in the liquid is at (0, 0, 0), the origin. Let Y (t) be the y-coordinate of
18. (Death Process with Immigration) Consider a population of size n, n ≥ 0, of a certain species in which individuals do not reproduce. However, new individuals immigrate into the population at a
17. Consider a population of a certain colonizing species. Suppose that each individual produces offspring at a Poisson rate of λ as long as it lives, and the time until the first individual arrives
16. In Springfield, Massachusetts, people drive their cars to a state inspection center for annual safety and emission certification at a Poisson rate of λ. For n ≥ 0, if there are n cars at the
15. There are m machines in a factory operating independently. Each machine works for a time period that is exponentially distributed with mean 1/µ. Then it breaks down. The time that it takes to
14. In a factory, there are m operating machines and s machines used as spares and ready to operate. The factory has k repairpersons, and each repairperson repairs one machine at a time. Suppose
13. Consider a parallel system consisting of two components denoted by 1 and 2. Such a system functions if and only if at least one of its components functions. Suppose that each component functions
12. Passengers arrive at a train station according to a Poisson process with rate λ and, independently, trains arrive at the same station according to another Poisson process, but with the same rate
11. On a given vacation day, Francesco either plays golf (activity 1) or tennis (activity 2). For i = 1, 2, let Xn = i, if Francesco devotes vacation day n to activity i. Suppose that{Xn : n = 1,
10. An urn contains 7 red, 11 blue, and 13 yellow balls. Carmela, Daniela, and Lucrezia play a game in which they take turns and draw balls from the urn, successively, at random and with replacement.
9. In a golfball production line, a golfball produced with no logo is called defective; all other golfballs are called good. A quality assurance engineer performing statistical process control

Showing 600 - 700 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