New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
business
probability and stochastic modeling

Fundamentals Of Probability With Stochastic Processes 3rd Edition Saeed Ghahramani - Solutions

32. Let Z be a standard normal random variable. Show that for x > 0, lim t→∞ P * Z > t + x t | Z ≥ t , = e−x . Hint: Use part (b) of Exercise 31.
31. (a) Prove that for all x > 0, 1 x √2π * 1 − 1 x2 , e−x2/2 < 1 − -(x) < 1 x √2π e−x2/2 . Hint: Integrate the following inequalities: (1 − 3y−4 )e−y2/2 < e−y2/2 < (1 + y−2 )e−y2/2 . (b) Use part (a) to prove that 1 − -(x) ∼ 1 x √2π e−x2/2 . That is, as x →
30. Prove that for some constant k, f (x) = ka−x2 , a ∈ (0,∞), is a normal probability density function.
29. 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 2500, is it fair to say that this algorithm is not accurate?
28. 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 approximate probability that this sample contains at least 18 light bulbs of each type?
27. 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 bestseller should the store stock every month so that, with a probability of more than 98%, it does not
26. Let X ∼ N (0, 1). Calculate the density function of Y = √|X|. B
25. Let X ∼ N (µ, σ2). Calculate the density function of Y = eX.
24. Let X ∼ N (0, σ2). Calculate the density function of Y = X2.
23. Let α ∈ (−∞,∞) and Z ∼ N (0, 1); find E(eαZ).
22. In a certain town the length of residence of a family in a home is normal with mean 80 months and variance 900. What is the probability that of 12 independent families, living on a certain street of that town, at least three will have lived there more than eight years?
21. The viscosity of a brand of motor oil is normal with mean 37 and standard deviation 10. What is the lowest possible viscosity for a specimen that has viscosity higher than at least 90% of that brand of motor oil?
20. Determine the value(s) of k for which the following is the probability density function of a normal random variable. f (x) = √ ke−k2x2−2kx−1 , −∞ < x < ∞.
19. Let X ∼ N (µ, σ2). Find the probability 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) = F 2 π e−2(x−1)2 .
17. (Investment) 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 would like to purchase enough shares of this stock to make at least $1000 profit with a
16. Suppose that lifetimes of light bulbs produced by a certain company are normal random variables with mean 1000 hours and standard deviation 100 hours. Suppose that lifetimes of light bulbs produced by a second company are normal random variables with mean 900 hours and standard deviation 150
15. Suppose that lifetimes of light bulbs produced by a certain company are normal random variables with mean 1000 hours and standard deviation 100 hours. Is this company correct when it claims that 95% of its light bulbs last at least 900 hours?
14. Let X ∼ N (µ, σ2). Prove that P ! |X − µ| > kσ " does not depend on µ or σ.
13. A number t is said to be the median of a continuous random variable X if P (X ≤ t) = P (X ≥ t) = 1/2. Calculate the median of the normal random variable with parameters µ and σ2.
12. Suppose that the scores on a certain manual dexterity test are normal with mean 12 and standard deviation 3. If eight randomly selected individuals take the test, what is the probability that none will make a score less than 14?
11. The amount of cereal in a box is normal with mean 16.5 ounces. If the packager is required to fill at least 90% of the cereal boxes with 16 or more ounces of cereal, what is the largest standard deviation for the amount of cereal in a box?
10. Suppose that the IQ of a randomly selected student from a university is normal with mean 110 and standard deviation 20. Determine the interval of values that is centered at the mean and includes 50% of the IQ’s of the students at that university.
9. The length of an aluminum-coated steel sheet manufactured by a certain factory is approximately normal with mean 75 centimeters and standard deviation 1 centimeter. Find the probability that a randomly selected sheet manufactured by this factory is between 74.5 and 75.8 centimeters.
8. Suppose that the distribution of the diastolic blood pressure is normal for a randomly selected person in a certain population is normal with mean 80 mm Hg and standard deviation 7 mm Hg. If people with diastolic blood pressures 95 or above are considered hypertensive and people with diastolic
7. The grades for a certain exam are normally distributed with mean 67 and variance 64. What percent of students get A(≥ 90), B(80 − 90), C(70 − 80), D(60 − 70), and F(< 60)?
6. The ages of subscribers to a certain newspaper are normally distributed with mean 35.5 years and standard deviation 4.8. What is the probability that the age of a random subscriber is (a) more than 35.5 years; (b) between 30 and 40 years?
5. Let X be a standard normal random variable. Calculate E(X cos X), E(sin X), and E * X 1 + X2 , .
4. Let Z be a standard normal random variable and α be a given constant. Find the real number x that maximizes P (x < Z < x + α).
3. 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.
2. 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 each year has 365 days.
1. 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?
17. Let Y be a random number from (0, 1). Let X be the second digit of √Y . Prove that for n = 0, 1, 2, . . . , 9, P (X = n)increases as n increases. This is remarkable because it shows that P (X = n), n = 1, 2, 3,... is not constant. That is, Y is uniform but X is not.
16. The 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 over the interval (−1, 4).
15. Let g be a nonnegative real-valued function on R that satisfies the relation # ∞ −∞ g(t) dt = 1. Show that if, for a random variable X, the random variable Y = # X −∞ g(t) dt is uniform, then g is the density function of X.
14. Let X be a continuous random variable with distribution function F. Prove that F (X) is uniformly distributed over (0, 1).
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 E 4 g(X)5 = θ 2.
12. Let X be a random number from (0, 1). Find the density functions of (a) Y = − ln(1 − X), and (b) Z = Xn.
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. B
10. Let θ be a random number between −π/2 and π/2. Find the probability density function of X = tan θ.
9. A farmer who has two pieces of lumber of lengths a and b (a
8. From the class of all triangles one is selected at random. What is the probability that it is obtuse? Hint: The largest angle of a triangle is less than 180 degrees but greater than or equal to 60 degrees.
7. A point is selected at random on a line segment of length $. What is the probability that none of the two segments is smaller than $/3?
6. A point is selected at random on a line segment of length $. What is the probability that the longer segment is at least twice as long as the shorter segment?
5. The radius of a sphere is a random number between 2 and 4. What is the expected value of its volume? What is the probability that its volume is at most 36π?
4. Suppose that b is a random number from the interval (−3, 3). What is the probability that the quadratic equation x2 + bx + 1 = 0 has at least one real root?
3. The time at which a bus arrives at a station is uniform over an interval (a,b) with mean 2:00 P.M. and standard deviation √12 minutes. Determine the values of a and b.
2. Suppose that 15 points are selected at random and independently from the interval (0, 1). How many of them can be expected to be greater than 3/4?
1. It takes a professor a random time between 20 and 27 minutes to walk from his home to school every day. If he has a class at 9:00 A.M. and he leaves home at 8:37 A.M., find the probability that he reaches his class on time.
11. The lifetime (in hours) of a light bulb manufactured by a certain company is a random variable with probability density function f (x) =    0 if x ≤ 500 5 × 105 x3 if x > 500.Suppose that, for all nonnegative real numbers a andb, the event that any light bulb lasts at
10. Let X be a continuous random variable with set of possible values{x : 0 < x < α} (where α < ∞), distribution functionF, and density function f . Using integration by parts, prove the following special case of Theorem 6.2. E(X) = E α 0 4 1 − F (t)5 dt.
9. Prove or disprove: If/n i=1 αi = 1, αi ≥ 0, ∀i, and $ fi %n i=1 is a sequence of density functions, then /n i=1 αifi is a probability density function.
8. Let F, the distribution of a random variable X, be defined by F (x) =    0 x < −1 1 2 + arcsin x π −1 ≤ x < 1 1 x ≥ 1, where arcsin x lies between −π/2 and π/2. Find f , the probability density function of X and E(X).
7. The probability density function of a continuous random variable X is f (x) = B 30x2(1 − x)2 if 0 < x < 1 0 otherwise. Find the probability density function of Y = X4.
6. Let X be a random variable with density function f (x) =    4x3/15 1 ≤ x ≤ 2 0 otherwise. Find the density functions of Y = eX, Z = X2, and W = (X − 1)2.
5. Does there exist a constant c for which the following is a density function? f (x) =    c 1 + x if x > 0 0 otherwise.
4. Let X be a random variable with density function f (x) = e−|x| 2 , −∞ < x < ∞. Find P (−2 < X < 1).
2. Let X be a continuous random variable with the probability density function f (x) =    2/x3 if x > 1 0 otherwise. Find E(X) and Var(X) if they exist.
1. Let X be a random number from (0, 1). Find the probability density function of Y = 1/X.
21. Let X be a continuous random variable with probability density function f . Show that if E(X) exists; that is, if # ∞ −∞ |x|f (x) dx < ∞, then lim x→−∞ xP (X ≤ x) = lim x→∞ xP (X > x) = 0.
20. Suppose that X is the lifetime of a randomly selected fan used in certain types of diesel engines. Let Y be a randomly selected competing fan for the same type of diesel engines manufactured by another company. To compare the lifetimes X and Y , it is not sufficient to compare E(X) and E(Y ).
19. Let X be the random variable introduced in Exercise 12. Applying the results of Exercise 17, calculate Var(X) .
18. Let X be a continuous random variable. Prove that .∞ n=1 P ! |X| ≥ n " ≤ E ! |X| " ≤ 1 +.∞ n=1 P ! |X| ≥ n " . These important inequalities show that E ! |X| " / < ∞ if and only if the series ∞ n=1 P ! |X| ≥ n " converges. Hint: By Exercise 17, E ! |X| " = E ∞ 0 P ! |X| > t"
17. Let X be a nonnegative random variable with distribution function F. Define I (t) = B 1 if X > t 0 otherwise. (a) Prove that # ∞ 0 I (t) dt = X. (b) By calculating the expected value of both sides of part (a), prove that E(X) = E ∞ 0 4 1 − F (t)5 dt. This is a special case of Theorem 6.2.
16. Let X be a continuous random variable with probability density function f (x). Determine the value of y for which E ! |X − y| " is minimum.
15. Let X be a continuous random variable with density function f . A number t is said to be the median of X if P (X ≤ t) = P (X ≥ t) = 1 2 .By Exercise 7, Section 6.1, X is symmetric about α if and only if for all x we have f (α − x) = f (α + x). Show that if X is symmetric about α, then
14. Let X be a continuous random variable with the probability density function f (x) =    1 π x sin x if 0 < x < π 0 otherwise. Prove that E(Xn+1 ) + (n + 1)(n + 2)E(Xn−1 ) = πn+1 .
13. For n ≥ 1, let Xn be a continuous random variable with the probability density function fn(x) =    cn xn+1 if x ≥ cn 0 otherwise. Xn’s are called Pareto random variables and are used to study income distributions. (a) Calculate cn, n ≥ 1. (b) Find E(Xn), n ≥ 1. (c)
12. Suppose that X, the interarrival time between two customers entering a certain postoffice, satisfies P (X > t) = αe−λt + βe−µt, t ≥ 0, where α + β = 1, α ≥ 0, β ≥ 0, λ > 0, µ > 0. Calculate the expected value of X. Hint: For a fast calculation, use Remark 6.4.
11. Let X be a random variable with the probability density function f (x) = 1 π(1 + x2) , −∞ < x < ∞. Prove that E ! |X| α" converges if 0 < α < 1 and diverges if α ≥ 1.
10. Let X be a random variable with probability density function f (x) = 1 2 e−|x| , −∞ < x < ∞. Calculate Var(X).
9. A right triangle has a hypotenuse of length 9. If the probability density function of one side’s length is given by f (x) =    x/6 if 2 < x < 4 0 otherwise, what is the expected value of the length of the other side?
7. Let the probability density function of tomorrow’s Celsius temperature be h. In terms of h, calculate the corresponding probability density function and its expectation for Fahrenheit temperature. Hint: Let C and F be tomorrow’s temperature in Celsius and Fahrenheit, respectively. Then F =
6. Let Y be a continuous random variable with probability distribution function F (y) = B e−k(α−y)/A −∞ < y ≤ α 1 y > α, where A, k, and α are positive constants. (Such distribution functions arise in the study of local computer network performance.) Find E(Y ).
4. A random variable X has the density function f (x) = B 3e−3x if 0 ≤ x < ∞ 0 otherwise. Calculate E(eX). 5. Find the expected value of a random variable X with the density function f (x) =    1 π √1 − x2 if −1 < x < 1 0 otherwise.
3. The mean and standard deviation of the lifetime of a car muffler manufactured by company A are 5 and 2 years, respectively. These quantities for car mufflers manufactured by company B are, respectively, 4 years and 18 months. Brian buys one muffler from company A and one from company B. That of
2. The time it takes for a student to finish an aptitude test (in hours) has the density function f (x) = B 6(x − 1)(2 − x) if 1 < x < 2 0 otherwise. Determine the mean and standard deviation of the time it takes for a randomly selected student to finish the aptitude test.
1. The distribution function for the duration of a certain soap opera (in tens of hours) is F (x) =    1 − 16 x2 if x ≥ 4 0 if x < 4. (a) Find E(X). (b) Show that Var(X) does not exist.
8. Let X be a random variable with the probability density function given by f (x) = B e−x if x ≥ 0 0 elsewhere. Let Y =    X if X ≤ 1 1/X if X > 1. Find the probability density function of Y .
7. Let X be a random variable with the density function f (x) = 1 π(1 + x2) , −∞ < x < ∞. (X is called a Cauchy random variable.) Find the density function of Z = arctan X.
6. Let f be the probability density function of a random variable X. In terms of f , calculate the probability density function of X2.
5. Let the probability density function of X be f (x) = B λe−λx if x ≥ 0 0 otherwise, for some λ > 0. Using the method of distribution functions, calculate the probability density function of Y = √3 X2.
4. Let X be a continuous random variable with the density function f (x) = B 3e−3x if x > 0 0 otherwise. Using the method of transformations, find the probability density function of Y = log2 X.
3. Let the density function of X be f (x) = B e−x if x > 0 0 elsewhere. Using the method of transformations, find the density functions of Y = X √X and Z = e−X.
2. Let X be a continuous random variable with distribution function F and density function f . Calculate the density function of the random variable Y = eX.
1. Let X be a continuous random variable with the density function f (x) =    1/4 if x ∈ (−2, 2) 0 otherwise. Using the method of distribution functions, find the probability density functions of Y = X3 and Z = X4.
11. The distribution function of a random variable X is given by F (x) = α + β arctan x 2 , −∞ < x < ∞. Determine the constants α and β and the density function of X.
10. Prove that if f and g are two probability density functions, then for α ≥ 0, β ≥ 0, and α + β = 1, αf + βg is also a probability density function.
9. Let X denote the lifetime of a radio, in years, manufactured by a certain company. The density function of X is given by f (x) =    1 15 e−x/15 if 0 ≤ x < ∞ 0 elsewhere. What is the probability that, of eight such radios, at least four last more than 15 years?
8. Suppose that the loss in a certain investment, in thousands of dollars, is a continuous random variable X that has a density function of the form f (x) = B k(2x − 3x2) −1 < x < 0 0 elsewhere. (a) Calculate the value of k. (b) Find the probability that the loss is at most $500.
7. Let X be a continuous random variable with density function f . We say that X is symmetric about α if for all x, P (X ≥ α + x) = P (X ≤ α − x). (a) Prove thatX is symmetric about α if and only if for all x, we have f (α−x) = f (α + x). (b) Let X be a continuous random variable with
6. Let X be a continuous random variable with density and distribution functions f and F, respectively. Assuming that α ∈ R is a point at which P (X ≤ α) < 1, prove that h(x) =    f (x) 1 − F (α) if x ≥ α 0 if x < α is also a probability density function.
5. The probability density function of a random variable X is given by f (x) =    c √1 − x2 if −1 < x < 1 0 elsewhere. (a) Calculate the value ofc. (b) Find the probability distribution function of X.
4. The lifetime of a tire selected randomly from a used tire shop is 10, 000X miles, where X is a random variable with the density function f (x) =    2/x2 if 1 < x < 2 0 elsewhere. (a) What percentage of the tires of this shop last fewer than 15,000 miles? (b) What percentage of those
3. The time it takes for a student to finish an aptitude test (in hours) has a density function of the form f (x) = B c(x − 1)(2 − x) if 1 < x < 2 0 elsewhere. (a) Determine the constantc. (b) Calculate the distribution function of the time it takes for a randomly selected student to finish the
2. The distribution function for the duration of a certain soap opera (in tens of hours) is F (x) =    1 − 16 x2 x ≥ 4 0 x < 4. (a) Calculate f , the probability density function of the soap opera. (b) Sketch the graphs of F and f . (c) What is the probability that the soap opera takes
1. When a certain car breaks down, the time that it takes to fix it (in hours) is a random variable with the density function f (x) = B ce−3x if 0 ≤ x < ∞ 0 otherwise. (a) Calculate the value ofc. (b) Find the probability that when this car breaks down, it takes at most 30 minutes to fix it.
27. Show that if all three of n, N, and D → ∞ so that n/N → 0, D/N converges to a small number, and nD/N → λ, then for all x, ; D x
26. A farmer, plagued by insects, seeks to attract birds to his property by distributing seeds over a wide area. Let λ be the average number of seeds per unit area, and suppose that the seeds are distributed in a way that the probability of having any number of seeds in a given area depends only
25. Suppose that n babies were born at a county hospital last week. Also suppose that the probability of a baby having blonde hair is p. If k of these n babies are blondes, what is the probability that the ith baby born is blonde?

Showing 2200 - 2300 of 6914

First
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved