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probability and stochastic modeling

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

4. A random sample of size n (n ≥ 1) is taken from a distribution with the following probability density function: f (x) = 1 2 e−|x| , −∞ < x < ∞. What is the probability that the sample mean is positive?
3. A random sample of size 24 is taken from a distribution with probability density function f (x) =    1 9 * x + 5 2 , 1 < x < 3 0 otherwise. Let X¯ be the sample mean. Approximate P (2 < 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
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
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 < ∞. Let Sn = X1 + X2 + · · · + Xn, X¯ n = Sn/n. Show that Sn grows at rate n. That is, lim n→∞
3. Let X be a nonnegative continuous random variable with probability density function f (x). Define Yn = B 1 if X > n 0 otherwise. Prove that Yn converges to 0 in probability.
2. Let {X1, X2,...} be a sequence of independent, identically distributed random variables with positive expected value. Show that for all M > 0, lim n→∞ P (X1 + X2 + · · · + Xn > M) = 1.
1. Let{X1, X2,...} be a sequence of nonnegative independent random variables and, for all i, suppose that the probability density function of Xi is f (x) = B 4x(1 − x) if 0 ≤ x ≤ 1 0 otherwise. Find lim n→∞ X1 + X2 + · · · + Xn n .
21. Let {x1, x2, . . . , xn} be a set of real numbers and define x¯ = 1 n .n i=1 xi, s2 = 1 n − 1 .n i=1 (xi − x)¯ 2 . Prove 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 = [
20. Let the probability density function of a random variable X be f (x) = xn n! e−x , x ≥ 0. Show that P (0 < X < 2n + 2) > n n + 1 . Hint: Note that # ∞ 0 xne−x dx = 0(n + 1) = n!. Use this to calculate E(X) and Var(X). Then apply Chebyshev’s inequality.
19. Let X be a random variable; show that for α > 1 and t > 0, P * X ≥ 1 t ln α , ≤ 1 α 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 4 (X − Y )2 5 = 0; then use Exercise 17.
17. Prove that if the random variables X and Y satisfy E 4 (X − Y )2 5 = 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(e kX) e kt .
15. Let X be a random variable with mean µ. Show that if E 4 (X − µ)2n 5 < ∞, then for α > 0, P ! |X − µ| ≥ α " ≤ 1 α2n E 4 (X − µ)2n5 .
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 pJ be the proportion of heads obtained. Determine the value of n for which pJestimates 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).
9. Suppose that X is a random variable with E(X) = Var(X) = µ. What does Chebyshev’s inequality say about P (X > 2µ)?
8. Show that for a nonnegative random variable X with mean µ, P (X ≥ 2µ) ≤ 1/2.
7. The waiting period from the time a book is ordered until 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 certain date, how early should she order the book?
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
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?
3. Let X be a nonnegative random variable with E(X) = 5 and E(X2) = 42. Find an upper bound for P (X ≥ 11) using (a) Markov’s inequality, (b) Chebyshev’s inequality.
2. Show that if for a nonnegative random variable X, P (X < 2) = 3/5, then E(X) ≥ 4/5.
1. According to the Bureau of Engraving and Printing, http://www.moneyfactory.com/document.cfm/18/106, December 10, 2003, the average life of a one-dollar Federal Reserve note is 22 months. Show that at most 37% of one-dollar bills last for at least five years.
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, p(x1, x2, . . . , xr) = n! x1! x2!··· xr! px1 1 px2 2 ··· pxr r , where x1 + x2 + · · · + xr = n, and p1 + p2 + · · · + pr = 1. Show that for k < r, X1 + X2 + · · · + Xk has a binomial
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. 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 11 2 years before the muffler of
16. 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
15. 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?
14. 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
13. 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 6, Section 10.5.
12. 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
11. 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?
10. 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.
9. 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
8. LetX, Y , andZ 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).
7. 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.
6. 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
5. Let X1, X2, . . . , Xn be n independent gamma random variables with parameters (r1, λ), (r2, λ), ... , (rn, λ), respectively. Use moment-generating functions to find the probability distribution function of X1 + X2 + · · · + Xn.
4. 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.
3. Let X1, X2, . . . , Xn be n independent exponential random variables with the identical mean 1/λ. Use moment-generating functions to find the probability distribution function of X1 + X2 + · · · + Xn.
2. 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).
1. Show that if X is a normal random variable with parameters (µ, σ2), then for α ∈ R, we have that MαX(t) = exp 4 αµt + (1/2)α2σ2t 2 5 .
26. 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 ,= j ≤ r. Hint: Note that by Remark 9.3, Xi and Xj are binomial random variables and the joint marginal probability mass
25. 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
24. 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.
23. Let X be a discrete random variable with probability mass function p(i) = 6 π2i2 , i = 1, 2, 3,... ; zero elsewhere. Show that the moment-generating function of X does not exist. Hint: Show that MX(t) is a divergent series on (0,∞). This implies that on no interval of the form (−δ, δ),
22. 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 probability distribution function; (b) the function MX(t) is an even function.
21. 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).
20. Let Z ∼ N (0, 1). Use MZ(t) = et2/2 to calculate E(Zn), where n is a positive integer. Hint: Use (11.2).
19. 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).
18. 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. Use (11.2).
17. For a random variable X, MX(t) = (1/81)(et + 2)4. Find P (X ≤ 2).
16. In each of the following cases MX(t), the moment-generating function of X, is given. Determine the distribution of X. (a) MX(t) = *1 4 et + 3 4 ,7 . (b) MX(t) = et /(2 − et ). (c) MX(t) = 4 2/(2 − t)] r. (d) MX(t) = exp 4 3(et − 1) 5 . Hint: Use Table 3 of the Appendix.
15. Prove that the function t/(1−t), t < 1, cannot be the moment-generating function of a random variable.
14. Suppose that the moment-generating function of X is given by MX(t) = et + e−t 6 + 2 3 , −∞ < t < ∞. Find E(Xr), r ≥ 1.
13. For a random variable X, MX(t) = 4 2/(2 − t)53 . Find E(X) and Var(X).
12. Let MX(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.
11. Suppose that the moment-generating function of a random variable X is given by MX(t) = 1 3 et + 4 15 e3t + 2 15 e4t + 4 15 e5t . Find the probability mass function of X.
10. Let MX(t) = (1/21) /6 n=1 nent . Find the probability mass function of X.
9. Let X be a geometric random variable with parameter p. Show that the momentgenerating function of X is given by MX(t) = pet 1 − qet , q = 1 − p, t < − ln q. Use MX(t) to find E(X) and Var(X).
8. Let X be a uniform random variable over the interval (a, b). Find the momentgenerating function of X.
7. (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).
6. Let X be a discrete random variable. Prove that E(Xn) = M(n) X (0).
5. Let X be a continuous random variable with the probability density function f (x) = 6x(1 − x), if 0 ≤ x ≤ 1, zero elsewhere. (a) Find MX(t). (b) Using MX(t), find E(X).
4. Let X be a continuous random variable with probability density function f (x) = 2x, if 0 ≤ x ≤ 1, zero elsewhere. Find the moment-generating function of X.
3. Let X be a discrete random variable with the probability mass function p(i) = 2 *1 3 ,i , i = 1, 2, 3,... ; zero elsewhere. Find MX(t) and E(X).
2. Let X be a random variable with probability density function f (x) =    1/4 if x ∈ (−1, 3) 0 otherwise. (a) Find MX(t), E(X), and Var(X). (b) Using MX(t), calculate E(X). Hint: Note that by the definition of derivative, M7 X(0) = lim h→0 MX(h) − MX(0) h .
1. Let X be a discrete random variable with probability mass function p(i) = 1/5, i = 1, 2, . . . , 5, zero elsewhere. Find MX(t).
16. Let {X1, X2, X3,...} be a sequence of independent and identically distributed exponential random variables with parameter λ. Let N be a geometric random variable with parameter p independent of {X1, X2, X3,...}. Find the distribution function of /N i=1 Xi.
14. Let the joint probability density function of X and Y be given by f (x, y) = B ye−y(1+x) if x > 0, y > 0 0 otherwise. (a) Show that E(X) does not exist. (b) Find E(X|Y ).
13. In terms of the means, variances, and the covariance of the random variables X and Y , find α and β for which E(Y − α − βX)2 is minimum. This is the method of least squares; it fits the “best” line y = α + βx to the distribution of Y .
12. Let the joint probability density function of X and Y be given by f (x, y) = B e−x if 0 < y < x < ∞ 0 elsewhere. (a) Find the marginal probability density functions of X and Y . (b) Determine the correlation coefficient of X and Y .
11. A random point (X, Y ) is selected from the rectangle [0, π/2] ×[ 0, 1]. What is the probability that it lies below the curve y = sin x?
10. Two green and two blue dice are rolled. If X and Y are the numbers of 6’s on the green and on the blue dice, respectively, calculate the correlation coefficient of |X − Y | and X + Y .
9. Two dice are rolled. The sum of the outcomes is denoted by X and the absolute value of their difference by Y . Calculate the covariance of X and Y . Are X and Y uncorrelated? Are they independent?
8. Let the joint probability mass function of discrete random variables X and Y be given by p(x, y) =    1 25(x2 + y2 ) if (x, y) = (1, 1), (1, 3), (2, 3) 0 otherwise.Find Cov(X, Y ).
6. Let the joint probability density function of X, Y , and Z be given by f (x, y, z) = B 8xyz if 0 < x < 1, 0 < y < 1, 0 < z < 1 0 otherwise. Find ρ(X, Y ), ρ(X, Z), and ρ(Y, Z).
3. Let the joint probability density function of random variables X and Y be f (x, y) =    3x3 + xy 3 if 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 0 elsewhere. Find E(X2 + 2XY ).
2. Let the probability density function of a random variable X be given by f (x) = B 2x − 2 if 1 < x < 2 0 elsewhere. Find E(X3 + 2X − 7).
1. In a commencement ceremony, for the dean of a college to present the diplomas of the graduates, a clerk piles the diplomas in the order that the students will walk on the stage. However, the clerk mixes the last 10 diplomas in some random order accidentally. Find the expected number of the last
10. Each year, a business loses some money because the market prices of some merchandise in its inventory decline. Let X be the amount of loss in a random year, in thousands of dollars, and suppose that the probability density function of X is given byOf all the years that the company loses at
9. Let X be a random variable with probability density functioni) Determine the probability density function of eX .(ii) Calculate E(eX). (2/3)x if 1 <
8. Let X be a continuous random variable with probability density functionFind E(X), Var(X), and E(cosX) COS I
7. Let F, the distribution of a random variable X, be defined bywhere arcsin x lies between −π/2 and π/2. Findf, the probability density function of X and E(X). F(x)= = 0 IN + arcsin x 1
6. Let X be a continuous random variable with probability density functionFind Var(X). f(x)= 0 COS T -/2x/2 otherwise.
5. At a local hardware store, the demand, per week, for propane gas, primarily for barbecue propane tanks, in 100’s of gallons, is a randomvariable with probability density function(a) How many gallons of propane gas must the hardware store carry, per week, so that the stock out probability for
4. There is a spring inside pendulum clocks that is an important component of its mechanism.Suppose that, for somec, the lifetime of such a spring, in tens of thousands of hours, is a random variable that has a probability density function of the form(i) Determine the constant c.(ii) Find the
3. An actuary has calculated that the probability density function of the cost to repair a vehicle after a certain type of car accident, in thousands of dollars, isThe actuary has also calculated that, for all auto insurance policies with the same deductible amountd, in 7.24% of the time, the cost
2. When Kara throws a dart at a dartboard, X, the distance, in centimeters, between the impact point of the dart and O, the center of the dartboard, is a continuous random variable with probability density functionIf the bull’s eye of the dartboard is a circle of diameter 6 centimeters centered
1. Let X be a strictly positive and continuous random variable with distribution function F and probability density functionf. Find the distribution and the probability density functions of Y = log2 X.
10. The lifetime (in hours) of a light bulb manufactured by a certain company is a random variable with probability density functionSuppose that, for all nonnegative real numbers a andb, the event that any light bulb lasts at least a hours is independent of the event that any other light bulb lasts

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