A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 8, Problem 8.8P
In Problem 8.7, suppose that it takes a random time, uniformly distributed over (0, .5), to replace a failed bulb. Approximate the
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Chapter 8 Solutions
A First Course in Probability
Ch. 8 - Suppose that X is a random variable with mean and...Ch. 8 - From past experience, a professor knows that the...Ch. 8 - Use the central limit theorem to solve part (c) of...Ch. 8 - Let X1,...,X20 be independent Poisson random...Ch. 8 - Fifty numbers are rounded off to the nearest...Ch. 8 - A die is continually rolled until the total sum of...Ch. 8 - A person has 100 light bulbs whose lifetimes are...Ch. 8 - In Problem 8.7, suppose that it takes a random...Ch. 8 - If X is a gamma random variable with parameters...Ch. 8 - Civil engineers believe that W, the amount of...
Ch. 8 - Many people believe that the daily change of price...Ch. 8 - We have 100 components that we will put in use in...Ch. 8 - Student scores on exams given by a certain...Ch. 8 - A certain component is critical to the operation...Ch. 8 - An insurance company has 10.000 automobile...Ch. 8 - A.J. has 20 jobs that she must do in sequence,...Ch. 8 - Redo Example 5b under the assumption that the...Ch. 8 - Repeat part (a) of Problem 8.2 when it is known...Ch. 8 - A lake contains 4 distinct types of fish. Suppose...Ch. 8 - If X is a nonne9ative random variable with mean...Ch. 8 - Let X be a nonnegative random variable. Prove that...Ch. 8 - Prob. 8.22PCh. 8 - Let X be a Poisson random variable with mean 20....Ch. 8 - Prob. 8.24PCh. 8 - If X has variance 2, then , the positive square...Ch. 8 - If X has mean and standard deviation , the ratio...Ch. 8 - Compute the measurement signal-to-noise ratio-that...Ch. 8 - Let Zn,n1, be a sequence of random variables and...Ch. 8 - Prob. 8.5TECh. 8 - Prob. 8.6TECh. 8 - Prob. 8.7TECh. 8 - Explain why a gamma random variable with...Ch. 8 - Prob. 8.9TECh. 8 - If X is a Poisson random variable with mean , show...Ch. 8 - Prob. 8.11TECh. 8 - Prob. 8.12TECh. 8 - Prob. 8.13TECh. 8 - Prob. 8.14TECh. 8 - The number of automobiles sold weekly at a certain...Ch. 8 - Prob. 8.2STPECh. 8 - If E[X]=75E[Y]=75Var(X)=10var(Y)=12cov(X,Y)=3 give...Ch. 8 - Prob. 8.4STPECh. 8 - Prob. 8.5STPECh. 8 - Prob. 8.6STPECh. 8 - Prob. 8.7STPECh. 8 - Prob. 8.8STPECh. 8 - Prob. 8.9STPECh. 8 - A tobacco company claims that the amount of...Ch. 8 - Prob. 8.11STPECh. 8 - Prob. 8.12STPECh. 8 - The strong law of large numbers states that with...Ch. 8 - Each new book donated to a library must be...
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- Total marks 14 4. Let X and Y be random variables on a probability space (N, F, P) that take values in [0, ∞). Assume that the joint density function of X and Y on [0, ∞) × [0, ∞) is given by f(x, y) = 2e-2x-y Find the probability P(0 ≤ X ≤ 1,0 ≤ y ≤ 2). (ii) spectively. [6 Marks] Find the the probability density function of X and Y, re- [5 Marks] 111) Are the X and Y independent? Justify your answer! [3 Marks]arrow_forwardTotal marks 17 4. Let (,,P) be a probability space and let X : → R be a ran- dom variable that has Gamma(2, 1) distribution, i.e., the distribution of the random variable X is the probability measure on ((0, ∞), B((0, ∞))) given by (i) dPx(x) = xex dx. Find the characteristic function of the random variable X. [8 Marks] (ii) Using the result of (i), calculate the first three moments of the random variable X, i.e., E(X") for n = 1, 2, 3. Using Markov's inequality involving E(X³), (iii) probability P(X > 10). [6 Marks] estimate the [3 Marks]arrow_forward1. There are 8 balls in an urn, of which 6 balls are red, 1 ball is blue and 1 ball is white. You draw a ball from the urn at random, note its colour, do not return the ball to the urn, and then draw a second ball, note its colour, do not return the ball to the urn, and finally draw a third ball, note its colour. (i) (Q, F, P). Describe the corresponding discrete probability space [7 Marks] (ii) Consider the following event, A: At least one of the first two balls is red.arrow_forward
- 3. Consider the following discrete probability space. Let = {aaa, bbb, ccc, abc, acb, bac, bca, cab, cba}, i.e., consists of 3-letter 'words' aaa, bbb, ccc, and all six possible 3-letter 'words' that have a single letter a, a single letter b, and a single letter c. The probability measure P is given by 1 P(w) = for each weΩ. 9 Consider the following events: A: the first letter of a 'word' is a, B: the second letter of a 'word' is a, C: the third letter of a 'word' is a. answer! Decide whether the statements bellow are true or false. Justify your (i) The events A, B, C are pairwise independent. (ii) The events A, B, C are independent. Total marks 7 [7 Marks]arrow_forwardLet X and Y have the following joint probability density function: fxy(x,y) =1/(x²²), for >>1, y>1 0, otherwise Let U = 5XY and V = 3 x. In all question parts below, give your answers to three decimal places (where appropriate). (a) The non-zero part of the joint probability density function of U and V is given by fu,v(u,v) = A√³uc for some constants A, B, C. Find the value of A. Answer: 5 Question 5 Answer saved Flag question Find the value of B. Answer: -1 Question 6 Answer saved P Flag question (b) The support of (U,V), namely the values of u and vthat correspond to the non-zero part of fu,v(u,v) given in part (a), is given by:arrow_forwardTotal marks 13. 3. There are three urns. Urn I contains 3 blue balls and 5 white balls; urn II contains 2 blue balls and 6 white balls; urn III contains 4 blue balls and 4 white balls. Rolling a dice, if 1 appears, we draw a ball from urn I; if 4 or 5 or 6 appears, we draw a ball from urn II; if 2, or 3 appears, we draw a ball from urn III. (i) What is the probability to draw a blue ball? [7 Marks] (ii) Assume that a blue ball is drawn. What is the probability that it came from Urn I? [6 Marks] Turn over. MA-252: Page 3 of 4arrow_forward
- 3. Consider the discrete probability space with the sample space = {a, b, c, d, e, f, g, h} and the probability measure P given by P(w) for each wEN. Consider the following events: A = {a, c, e, g}, B = {b, c, d, e}, C = = {a, b, d, g}. Decide whether the statements bellow are true or false. Justify your answer! (i) The events A, B, C are pairwise independent. (ii) The events A, B, C are independent. Total marks 6 [6 Marks]arrow_forward2. space Consider the discrete probability space (N, F, P) with the sample N = {W1 W2 W3 W4 W5, W6, W7, W8, W9, W10, W11, W12}, is the power of 2, and the probability measure P is given by 1 P(wi) for each i = 1, 12. 12 Consider the following events: A = {W1, W3, W5, W7, W9, W11}, C = B = {W1, WA, W7, W8, W9, W12}, = {W3, WA, W5, W6, W9, W12}. Decide whether the statements bellow are true or false. Justify your answer! (i) The events A, B, C are pairwise independent. [5 Marks] Total marks 8 (ii) The events A, B, C are independent. [3 Marks]arrow_forwardshould my answer be 2.632 or -2.632?arrow_forward
- 2. Alice has three unfair coins. The first coin shows heads with proba- bility 0.3 and tails with probability 0.7. The second coin shows heads with probability 0.9 and tails with probability 0.1. The third coin shows heads with probability 0.4 and tails with probability 0.6. First, Alice throws a dice. If the dice yields a 1 or 2, Alice tosses the first coin. If the dice yields a 3, Alice tosses the second coin. If the dice yields a 4 or 5, or 6, Alice tosses the third coin. (i) What is the probability of the following event: as a result of throwing the dice and then tossing a coin, tails appear. [7 Marks] Assume a coin shows tails. What is the probability that Alice (ii) tossed (a) the first coin; (b) the second coin; Total marks 11 (c) the third coin? [4 Marks]arrow_forwardIn a Poisson process with A = 500 arrivals per day, find the approximate probability that the thousandth arrival comes between 47.5 and 48.5 hours after the start of observations.arrow_forwardSuppose you generate 300 independent observations X1, ..., X300 from a Uniform(0,3) distribution. a) What is the approximate distribution of X? b) What is the approximate probability that X is less than 1.57? c) What is the approximate probability that the sum S300 = X₁++X300 is between 442 and 483?arrow_forward
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