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probability and stochastic modeling

Fundamentals Of Probability With Stochastic Processes 4th Edition Saeed Ghahramani - Solutions

10. Consider a distribution with mean μ and probability density functionDetermine the values of n for which the probability is at least 0.98 that the mean of a random sample of size n from the population is within ±0.07 of μ? 1 4x6 f(x) = x In(3/2) elsewhere.
9. Suppose that, whenever invited to a party, the probability that a person attends with his or her guest is 1/3, attends alone is 1/3, and does not attend is 1/3. A company has invited all 300 of its employees and their guests to a Christmas party. What is the probability that at least 320 will
8. A physical quantity is measured 50 times, and the average of these measurements is taken as the result. If each measurement has a random error uniformly distributed over(−1, 1), what is the probability that our result differs from the actual value by less than 0.25?
7. Carla has invested $5000 in a stock. Suppose that, each month, the investment increases by $377 with probability 0.49 and decreases by $315 with probability 0.51. If the changes in stock prices in different months are independent, what is the probability that at the end of year 10, the value of
6. Each time that Jim charges an item to his credit card, he rounds the amount to the nearest dollar in his records. If he has used his credit card 300 times in the last 12 months, what is the probability that his record differs from the total expenditure by, atmost, 10 dollars?
5. Let X1,X2, . . . ,Xn be independent and identically distributed random variables, and let Sn = X1 + X2 + · · · + Xn. For large n, what is the approximate probability that Sn is between E(Sn) − σSn and E(Sn) + σSn?
4. A random sample of size n (n ≥ 1) is taken from a distribution with the following probability density function:What is the probability that the sample mean is positive? f(x) = e1, -x
3. A randomsample of size 24 is taken from a distribution with probability density functionLet ¯X be the sample mean. Approximate P(2 f(x) = 0 1 < x
2. For the scores on an achievement test given to a certain population of students, the expected value is 500 and the standard deviation is 100. Let ¯X be the mean of the scores of a random sample of 35 students from the population. Estimate P(460 < ¯X < 540).
1. What is the probability that the average of 150 random points from the interval (0, 1) is within 0.02 of the midpoint of the interval?
A biologist wants to estimate ℓ, the life expectancy of a certain type of insect. To do so, he takes a sample of size n and measures the lifetime from birth to death of each insect. Then he finds the average of these numbers. If he believes that the lifetimes of these insects are independent
If 20 random numbers are selected independently from the interval (0, 1), what is the approximate probability that the sum of these numbers is at least eight?
The time it takes for a student to finish an aptitude test (in hours) has the probability density functionApproximate the probability that the average length of time it takes for a random sample of 15 students to complete the test is less than 1 hour and 25 minutes. (6(x-1)(2-x) if1
The lifetime of a TV tube (in years) is an exponential random variable with mean 10. What is the probability that the average lifetime of a random sample of 36 TV tubes is at least 10.5?
2. For n ≥ 1, let Xn be a binomial random variable with parameters n and p. Since Xn is the number of successes in n independently performed Bernoulli trials, Xn is the fraction of successes in the first n trials. Show that Xn converges to p in probability.That is, for all ε > 0, 11-00 n lim
1. Let X1, X2, . . . , Xn be independent random numbers from the interval (a, b), a Find the value of α for whichwith probability 1. lim = n-xx X+X++X/ n
7. For a positive integer n, let τ (n) = (2k, i), where i is the remainder when we divide n by 2k, the largest possible power of 2. For example, τ (10) = (23, 2), τ (12) = (23, 4),τ (19) = (24, 3), and τ (69) = (26, 5). In an experiment a point is selected at random from [0, 1]. For n ≥ 1,
6. In Example 11.21, suppose that at t = 0 the bank is not free, there are m > 0 customers waiting in a queue to be served, and a customer is being served. Show that, with probability 1, eventually, for some period, the bank will be empty of customers.
5. Suppose that in Example 11.21 rather than customers being served in their arrival order, they are served on a last-come, first-served basis, or they are simply served in a random order. Show that if λ < μ, then still with probability 1, eventually, for some period, the bank will be empty of
4. Let {X1,X2, . . .} be a sequence of independent, identically distributed random variables.In other words, for all n, let X1,X2, . . . ,Xn be a random sample from a distribution with mean μ lim P(n(-) S n ( + c)) = 1.
3. Let X be a nonnegative continuous random variable with probability density function f(x). DefineProve that Yn converges to 0 in probability. Yn if X > n otherwise.
2. Let {X1,X2, . . .} be a sequence of independent, identically distributed random variables with positive expected value. Show that for allM > 0, lim P(X1+X2++Xn>M) = 1. x+2
1. Let {X1,X2, . . .} be a sequence of nonnegative independent random variables and, for all i, suppose that the probability density function of Xi isFind (6x(1-x) if 0x1 f(x) = 0 otherwise.
Suppose that there exist N families on the earth and that the maximum number of children a family has isc. Let αj , j = 0, 1, . . . , c, Pc j=0 αj = 1, be the fraction of families with j children. Determine the fraction of the children in the world who are from a family with k (k = 1, 2, . . .
Suppose that an “immortal monkey” is constantly typing on a word processor that is not breakable, lasts forever, and has infinite memory. Suppose that the keyboard of the word processor hasm−1 keys, a space bar for blank spaces, and separate keys for different symbols. If each time the monkey
At a large international airport, a currency exchange bank with only one teller is open 24 hours a day, 7 days a week. Suppose that at some time t = 0, the bank is free of customers and new customers arrive at random times T1, T1 + T2, T1 + T2 + T3,. . . , where T1, T2, T3, . . . are identically
2. A fair die is tossed 120 times. Let Y be the sum of the outcomes. Show that 155 P(330 Y510) 0.96. 162
1. Dr. O’Neill is a proficient, caring, and popular physician. When a patient calls for a non-urgent appointment, the number of days before he can see the patient is a random variable with mean 10 days and standard deviation 3 days. (That includes the day the patient calls, but not the day of the
21. Let {x1, x2, . . . , xn} be a set of real numbers and defineProve that at least a fraction 1 − 1/k2 of the xi’s are between ¯x − ks and ¯x + ks.Sketch of a Proof: Let N be the number of x1, x2, . . . , xn that fall in A = [¯x − ks, ¯x + ks]. ThenThis gives (N − 1)/(n − 1) ≥ 1
20. Let the probability density function of a random variable X beShow thatUse this to calculate E(X) and Var(X). Then apply Chebyshev’s inequality. f(x) = x 0. n!
19. Let X be a random variable; show that for α > 1 and t > 0, t P(x >= lna) - Mx(t).
18. Let X and Y be two randomly selected numbers from the set of positive integers{1, 2, . . . , n}. Prove that ρ(X, Y ) = 1 if and only if X = Y with probability 1.Hint: First prove that E(X − Y )2= 0; then use Exercise 17.
17. Prove that if the random variables X and Y satisfy E(X − Y )2= 0, then with probability 1, X = Y .
16. Let X be a random variable and k be a constant. Prove that P(X > t) E(ekx) ekt
15. Let X be a random variable with mean μ. Show that if E(X − μ)2nα > 0, P(X - | ) E[(X )n]. Q2n -
14. For a coin, p, the probability of heads is unknown. To estimate p, for some n, we flip the coin n times independently. Let bp be the proportion of heads obtained. Determine the value of n for which bp estimates p within ±0.05 with a probability of at least 0.94.
13. To determine p, the proportion of time that an airline operator is busy answering customers, a supervisor observes the operator at times selected randomly and independently from other observed times. Let Xi = 1 if the ith time the operator is observed, he is busy; let Xi = 0 otherwise. For
12. For a distribution, the mean of a random sample is taken as estimation of the expected value of the distribution. How large should the sample size be so that, with a probability of at least 0.98, the error of estimation is less than 2 standard deviations of the distribution?
11. The mean IQ of a randomly selected student from a specific university is μ; its variance is 150. A psychologist wants to estimate μ. To do so, for some n, she takes a sample of size n of the students at random and independently and measures their IQ’s. Then she finds the average of these
10. From a distribution with mean 42 and variance 60, a random sample of size 25 is taken.Let ¯X be the mean of the sample. Show that the probability is at least 0.85 that ¯X ∈ (38, 46).
7. The number of days from the time a book is ordered until it is received (including the day the book is ordered and the day that it is received) is a random variable with mean seven days and standard deviation two days. If Helen wants to be 95% sure that she receives a book by a certain date, how
6. The average IQ score on a certain campus is 110. If the variance of these scores is 15, what can be said about the percentage of students with an IQ above 140?
5. Suppose that the average number of accidents at an intersection is two per day.(a) Use Markov’s inequality to find a bound for the probability that at least five accidents will occur tomorrow.(b) Using Poisson random variables, calculate the probability that at least five accidents will occur
4. The average and standard deviation of lifetimes of light bulbs manufactured by a certain factory are, respectively, 800 hours and 50 hours.What can be said about the probability that a random light bulb lasts, at most, 700 hours?
1. According to the Board of Governors of the Federal Reserve System, October 10, 2017, https://www.federalreserve.gov/faqs/how-long-is-the-life-span-of-us-paper-money.htm, the average life of a one-dollar Federal Reserve note is 5.8 years. Show that at most 39%of one-dollar bills last for at least
For a coin, p, the probability of heads is unknown. To estimate p, we flip the coin 3000 times and let bp be the fraction of times it lands heads up. Show that the probability is at least 0.90 that bp estimates p within ±0.03.
To estimate the percentage of defective nails manufactured by a specific company, a quality control officer decides to take a random sample of nails produced by the company and calculate the fraction of those that are defective. If the officer wants to be sure, with a probability of at least 98%,
A biologist wants to estimate ℓ, the life expectancy of a certain type of insect. To do so, he takes a sample of size n and measures the lifetime from birth to death of each insect. Then he finds the average of these numbers. If he believes that the lifetimes of these insects are independent
For the scores on an achievement test given to a certain population of students, the expected value is 500 and the standard deviation is 100. Let ¯X be the mean of the scores of a random sample of 10 students. Find a lower bound for P(460 < ¯X < 540).
A blind will fit Myra’s bedroom’s window if its width is between 41.5 and 42.5 inches.Myra buys a blind from a store that has 30 such blinds.What can be said about the probability that it fits her window if the average of the widths of the blinds is 42 inches with standard deviation 0.25?
Suppose that, on average, a post office handles 10,000 letters a day with a variance of 2000. What can be said about the probability that this post office will handle between 8000 and 12,000 letters tomorrow?
A post office, on average, handles 10,000 letters per day. What can be said about the probability that it will handle (a) at least 15,000 letters tomorrow; (b) fewer than 15,000 letters tomorrow?
2. Let Z1, Z2, . . . , Zn be independent standard normal random variables. For what value of α, α > 0, is α(Z1 + Z2 + · · · + Zn) also a standard normal random variable?
1. Suppose that aMassachusetts-based insurance company insures residential properties in Boston, Worcester, and Springfield. Suppose that, during a given year, X1, X2, and X3 are the losses of this company due to house fires in the aforementioned areas, respectively.If, in millions of dollars, X1
20. Kim is at a train station, waiting to make a phone call. Two public telephone booths, next to each other, are occupied by two callers, and 11 persons are waiting in a single line ahead of Kim to call. If the duration of each telephone call is an exponential random variable with λ = 1/3, what
19. Let the joint probability mass function of X1,X2, . . . ,Xr be multinomial, that is,where x1 + x2 + · · · + xr = n, and p1 + p2 + · · · + pr = 1. Show that for k n! P(x1, x2,...,x) =
18. An elevator can carry up to 3500 pounds. The manufacturer has included a safety margin of 500 pounds and lists the capacity as 3000 pounds. The building’s management seeks to avoid accidents by limiting the number of passengers on the elevator. If the weight of the passengers using the
17. On a certain day, the first customer of a single-teller bank arrives at 9:00 A.M. when the bank opens. The second customer arrives 10 minutes later at 9:10 A.M. Suppose that the service times of customers are independent exponential random variables each with mean 10 minutes, customers wait in
16. Suppose that car mufflers last random times that are normally distributed with mean 3 years and standard deviation 1 year. If a certain family buys two new cars at the same time, what is the probability that (a) they should change the muffler of one car at least 1*1/2 years before the muffler
15. The distributions of the grades of the students of probability and calculus at a certain university are N(65, 418) and N(72, 448), respectively. Dr. Olwell teaches a calculus section with 28 and a probability section with 22 students. What is the probability that the difference between the
14. The capacity of an elevator is 2700 pounds. If the weight of a random athlete is normal with mean 225 pounds and standard deviation 25, what is the probability that the elevator can safely carry 12 random athletes?
13. Let X be the height of a man, and let Y be the height of his daughter (both in inches).Suppose that the joint probability density function of X and Y is bivariate normal with the following parameters: μX = 71, μY = 60, σX = 3, σY = 2.7, and ρ = 0.45. Find the probability that the man is at
12. Let the joint probability density function of X and Y be bivariate normal. Prove that any linear combination of X and Y, αX + βY, is a normal random variable.Hint: Use Theorem 11.7 and the result of Exercise 7, Section 10.5.
11. Vicki owns two department stores. Delinquent charge accounts at store 1 show a normal distribution, with mean $90 and standard deviation $30, whereas at store 2 they show a normal distribution with mean $100 and standard deviation $50. If 10 delinquent accounts are selected randomly at store 1
10. The distribution of the IQ of a randomly selected student from a certain college is N(110, 16). What is the probability that the average of the IQ’s of 10 randomly selected students from this college is at least 112?
9. Let X ∼ N(1, 2) and Y ∼ N(4, 7) be independent random variables. Find the probability of the following events: (a) X +Y > 0, (b) X − Y < 2, (c) 3X + 4Y > 20.
8. Mr. Watkins is at a train station, waiting to make a phone call. There is only one public telephone booth, and it is being used by someone. Another person ahead of Mr. Watkins is also waiting to call. If the duration of each telephone call is an exponential random variable with λ = 1/8, find
7. Let X, Y, and Z be three independent Poisson random variables with parameters λ1,λ2, and λ3, respectively. For y = 0, 1, 2, . . . , t, calculate P(Y = y | X+Y +Z = t).
6. Let X and Y be independent binomial random variables with parameters (n, p) and(m, p), respectively. Calculate P(X = i | X + Y = j) and interpret the result.
5. The probability is 0.15 that a bottle of a certain soda is underfilled, independent of the amount of soda in other bottles. If machine one fills 100 bottles and machine two fills 80 bottles of this soda per hour, what is the probability that tomorrow, between 10:00 A.M.and 11:00 A.M., both of
4. Let X1,X2, . . . ,Xn be n independent gamma random variables with parameters(r1, λ), (r2, λ), . . . , (rn, λ), respectively. Use moment-generating functions to show that the distribution of X1+X2+· · ·+Xn is gamma with parameters (r1+r2+· · ·+rn, λ).
3. Using moment-generating functions, show that the sum of n independent negative binomial random variables with parameters (r1, p), (r2, p), . . . , (rn, p) is negative binomial with parameters (r, p), r = r1 + r2 + · · · + rn.
2. Let X1,X2, . . . ,Xn be n independent exponential random variables with the identical mean 1/λ. Use moment-generating functions to show that the distribution of X1+X2+· · · + Xn is gamma with parameters n and λ.
1. Let X1,X2, . . . ,Xn be independent geometric random variables each with parameter p. Using moment-generating functions, prove that X1 + X2 + · · · + Xn is negative binomial with parameters (n, p).
Office fire insurance policies by a certain company have a $1000 deductible.The company has received three claims, independent of each other, for damages caused by office fire. If reconstruction expenses for such claims are exponentially distributed, each with mean $45,000, what is the probability
Suppose that the distribution of students’ grades in a probability test is normal, with mean 72 and variance 25.(a) What is the probability that the average grade of such a probability class with 25 students is 75 or more?(b) If a professor teaches two different sections of this course, each
4. Suppose that for a random variable X, E(Xn) = 2n, n = 1, 2, 3, . . .. Calculate the moment-generating function and the probability mass function of X. Hint: Note that = e, < x < . 0
3. For some α > 0, the moment-generating function of X is given byFor what value of α does Var(X) = 279/64? 3 Mx(t) = eat+2at 8 + 1 IN e3at 2
2. Can the functionbe the moment-generating function of a random variable X? F 5 f (t) = (e + ) 3
1. Let X be a binomial random variable with parameters n and p. Using momentgenerating functions, find the probability mass function of Y = n − X.
27. Let the joint probability mass function of X1, X2, . . . , Xr be multinomial with parameters n and p1, p2, . . . , pr (p1 + p2 + · · · + pr = 1). Find ρ(Xi,Xj), 1 ≤ i 6= j ≤ r.Hint: Note that by Remark 9.4,Xi andXj are binomial randomvariables and the joint marginal probability mass
26. Suppose that A dollars are invested in a bank that pays interest at a rate of X per year, where X is a random variable.(a) Show that if a year is divided into k equal periods, and the bank pays interest at the end of each of these k periods, then after n such periods, with probability 1, the
25. Suppose that ∀n ≥ 1, the nth moment of a random variable X, is given by E(Xn) =(n + 1)! 2n. Find the distribution of X.
24. Let X be a discrete random variable with probability mass functionShow that the moment-generating function of X does not exist.Hint: Show thatMX(t) is a divergent series on (0,∞). This implies that on no interval of the form (−δ, δ), δ > 0, MX(t) exists. p(i) = 6 = 22 i=1,2,3,... zero
23. Let X be a continuous random variable whose probability density function f is even;that is, f(−x) = f(x), ∀x. Prove that (a) the random variables X and −X have the same distribution function; (b) the functionMX(t) is an even function.
22. Let X be a gamma random variable with parameters r and λ. Derive a formula for MX(t), and use it to calculate E(X) and Var(X).
21. Let Z ∼ N(0, 1). UseMZ(t) = et2/2 to calculate E(Zn), where n is a positive integer.Hint: Use (11.2).
20. Let X be a uniform random variable over (0, 1). Let a and b be two positive numbers.Using moment-generating functions, show that Y = aX + b is uniformly distributed over (b, a + b)
19. For a random variable X, MX(t) = (1/81)(et + 2)4. Find P(X ≤ 2).
18. In each of the following cases MX(t), the moment-generating function of X, is given.Determine the distribution of X. 37 (a) Mx(t) = (e+). (b) Mx(t) e/(2- e). (c) Mx(t) = [2/(2- t)]". (d) Mx(t) = exp [3(et - 1)].
17. According to an actuary, the claim size for a certain brand of luxury car when it gets involved in an accident is a continuous random variable with moment-generating functionFind the expected value and the standard deviation of the claim size for such a car involved in an accident and selected
16. Suppose that the moment-generating function of X is given byFind E(Xr), r ≥ 1. Mx(t) = et + e-t 6 + 213 3' -
15. For a random variable X,MX(t) =2/(2 − t)3. Find E(X) and Var(X).
14. LetMX(t) = 1/(1−t), t < 1 be the moment-generating function of a random variable X. Find the moment-generating function of the random variable Y = 2X + 1.
13. Suppose that the moment-generating function of a random variable X is given byFind the probability mass function of X. Mx(t) === + 3 15 e3t+ 2 4t 15 15
12. LetMX(t) = (1/21)P6 n=1 nent. Find the probability mass function of X.
11. Let X be a geometric random variable with parameter p. Show that the momentgenerating function of X is given byUseMX(t) to find E(X) and Var(X). pet Mx(t) 1 - get' q=1-p, t
10. Let X be a uniform random variable over the interval (a, b). Find the momentgenerating function of X.
9. (a) Find MX(t), the moment-generating function of a Poisson random variable X with parameter λ.(b) Use MX(t) to find E(X) and Var(X).
8. Let X be a discrete random variable. Prove that E(Xn) = M(n)X (0).

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