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a. Let X 1 have a chi-squared distribution with parameter v 1 (see Section 4.4), and let X 2 be independent of X 1 and have a chi-squared distribution with parameter v 2 . Use the technique of Example 5.22 to show that X 1 + X 2 has a chi-squared distribution with parameter v 1 + v 2 . b. In Exercise 71 of Chapter 4, you were asked to show that if Z is a standard normal rv, then Z 2 has a chi-squared distribution with v = 1. Let Z 1 , Z 2 , . . . , Zn be n independent standard normal rv’s. What is the distribution of Z 1 2 +…+ Zn 2 ? Justify your answer. c. Let X 1 , … , X n be a random sample from a normal distribution with mean µ and variance σ 2 . What is the distribution of the sum Y = ∑ i = 1 n [ ( X i − μ ) / σ ] 2 ? Justify your answer.

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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809
BuyFind

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

Solutions

Chapter
Section
Chapter 5, Problem 91SE
Textbook Problem

a. Let X1 have a chi-squared distribution with parameter v1 (see Section 4.4), and let X2 be independent of X1 and have a chi-squared distribution with parameter v2. Use the technique of Example 5.22 to show that X1 + X2 has a chi-squared distribution with parameter v1 + v2.

b. In Exercise 71 of Chapter 4, you were asked to show that if Z is a standard normal rv, then Z2 has a chi-squared distribution with v = 1. Let Z1, Z2, . . . , Zn  be n independent standard normal rv’s. What is the distribution of Z12 +…+ Zn2 ? Justify your answer.

c. Let X1, … , Xn be a random sample from a normal distribution with mean µ and variance σ2. What is the distribution of the sum Y = i = 1 n [ ( X i μ ) / σ ] 2 ? Justify your answer.

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Chapter 5 Solutions

Probability and Statistics for Engineering and the Sciences
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Ch. 5.1 - Two different professors have just submitted final...Ch. 5.1 - Two components of a minicomputer have the...Ch. 5.1 - You have two lightbulbs for a particular lamp. Let...Ch. 5.1 - Suppose that you have ten lightbulbs, that the...Ch. 5.1 - Consider a system consisting of three components...Ch. 5.1 - a. For f(x1, x2, x3) as given in Example 5.10,...Ch. 5.1 - An ecologist wishes to select a point inside a...Ch. 5.1 - Refer to Exercise 1 and answer the following...Ch. 5.1 - The joint pdf of pressures for right and left...Ch. 5.1 - Let X1, X2, X3, X4, X5, and X6 denote the numbers...Ch. 5.1 - Let X1, X2, and X3 be the lifetimes of components...Ch. 5.2 - An instructor has given a short quiz consisting of...Ch. 5.2 - The difference between the number of customers in...Ch. 5.2 - Six individuals, including A and B, take seats...Ch. 5.2 - A surveyor wishes to lay out a square region with...Ch. 5.2 - Consider a small ferry that can accommodate cars...Ch. 5.2 - Annie and Alvie have agreed to meet for lunch...Ch. 5.2 - Show that if X and Y are independent rvs, then...Ch. 5.2 - Compute the correlation coefficient for X and Y...Ch. 5.2 - a. Compute the covariance for X and Y in Exercise...Ch. 5.2 - a. Compute the covariance between X and Y in...Ch. 5.2 - Reconsider the minicomputer component lifetimes X...Ch. 5.2 - Use the result of Exercise 28 to show that when X...Ch. 5.2 - a. Recalling the definition of 2 for a single rv...Ch. 5.2 - a. Use the rules of expected value to show that...Ch. 5.2 - Show that if Y = aX + b (a 0), then Corr(X, Y)...Ch. 5.3 - A particular brand of dishwasher soap is sold in...Ch. 5.3 - There are two traffic lights on a commuters route...Ch. 5.3 - It is known that 80% of all brand A external hard...Ch. 5.3 - A box contains ten sealed envelopes numbered 1, ....Ch. 5.3 - Let X be the number of packages being mailed by a...Ch. 5.3 - A company maintains three offices in a certain...Ch. 5.3 - Suppose the amount of liquid dispensed by a...Ch. 5.4 - Youngs modulus is a quantitative measure of...Ch. 5.4 - Refer to Exercise 46. 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Calculate the covariance...Ch. 5.5 - Suppose your waiting time for a bus in the morning...Ch. 5.5 - Suppose that when the pH of a certain chemical...Ch. 5.5 - If two loads are applied to a cantilever beam as...Ch. 5.5 - One piece of PVC pipe is to be inserted inside...Ch. 5.5 - Two airplanes are flying in the same direction in...Ch. 5.5 - Three different roads feed into a particular...Ch. 5.5 - Consider a random sample of size n from a...Ch. 5.5 - In Exercise 66, the weight of the beam itself...Ch. 5.5 - I have three errands to take care of in the...Ch. 5.5 - Suppose the expected tensile strength of type-A...Ch. 5.5 - In an area having sandy soil, 50 small trees of a...Ch. 5 - A restaurant serves three fixed-price dinners...Ch. 5 - In cost estimation, the total cost of a project is...Ch. 5 - A health-food store stocks two different brands of...Ch. 5 - According to the article Reliability Evaluation of...Ch. 5 - Suppose that for a certain individual, calorie...Ch. 5 - The mean weight of luggage checked by a randomly...Ch. 5 - We have seen that if E(X1) = E(X2) = =E(Xn) = ,...Ch. 5 - Suppose the proportion of rural voters in a...Ch. 5 - Let denote the true pH of a chemical compound. 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