Chapter 5, Problem 91SE

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9th Edition

Jay L. Devore

ISBN: 9781305251809

Textbook Problem

**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

**a.**

To determine

Prove that

Explanation

**Given info:**

*Reference- Example 5.22:* The random variables

**Calculation:**

The probability density function (pdf) of a random variable *X* having a chi-squared distribution with parameter

Denote

Make the transformations:

The Jacobian determinant of the transformation is:

**b.**

To determine

Find the distribution of

**c.**

To determine

Find the distribution of the sum

Sect-5.1 P-1ESect-5.1 P-2ESect-5.1 P-3ESect-5.1 P-4ESect-5.1 P-5ESect-5.1 P-6ESect-5.1 P-7ESect-5.1 P-8ESect-5.1 P-9ESect-5.1 P-10E

Sect-5.1 P-11ESect-5.1 P-12ESect-5.1 P-13ESect-5.1 P-14ESect-5.1 P-15ESect-5.1 P-16ESect-5.1 P-17ESect-5.1 P-18ESect-5.1 P-19ESect-5.1 P-20ESect-5.1 P-21ESect-5.2 P-22ESect-5.2 P-23ESect-5.2 P-24ESect-5.2 P-25ESect-5.2 P-26ESect-5.2 P-27ESect-5.2 P-28ESect-5.2 P-29ESect-5.2 P-30ESect-5.2 P-31ESect-5.2 P-32ESect-5.2 P-33ESect-5.2 P-34ESect-5.2 P-35ESect-5.2 P-36ESect-5.3 P-37ESect-5.3 P-38ESect-5.3 P-39ESect-5.3 P-40ESect-5.3 P-41ESect-5.3 P-42ESect-5.3 P-43ESect-5.4 P-46ESect-5.4 P-47ESect-5.4 P-48ESect-5.4 P-49ESect-5.4 P-50ESect-5.4 P-51ESect-5.4 P-52ESect-5.4 P-53ESect-5.4 P-54ESect-5.4 P-55ESect-5.4 P-56ESect-5.4 P-57ESect-5.5 P-58ESect-5.5 P-59ESect-5.5 P-60ESect-5.5 P-61ESect-5.5 P-62ESect-5.5 P-63ESect-5.5 P-64ESect-5.5 P-65ESect-5.5 P-66ESect-5.5 P-67ESect-5.5 P-68ESect-5.5 P-69ESect-5.5 P-70ESect-5.5 P-71ESect-5.5 P-72ESect-5.5 P-73ESect-5.5 P-74ECh-5 P-75SECh-5 P-76SECh-5 P-77SECh-5 P-78SECh-5 P-79SECh-5 P-80SECh-5 P-81SECh-5 P-82SECh-5 P-83SECh-5 P-84SECh-5 P-85SECh-5 P-86SECh-5 P-87SECh-5 P-88SECh-5 P-89SECh-5 P-90SECh-5 P-91SECh-5 P-92SECh-5 P-93SECh-5 P-94SECh-5 P-95SECh-5 P-96SECh-5 P-97SECh-5 P-98SE

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