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In Exercise 5.42, the number of defects per yard in a

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer ISBN: 9780495110811 47

Solution for problem 136E Chapter 5

Mathematical Statistics with Applications | 7th Edition

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Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

Mathematical Statistics with Applications | 7th Edition

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Problem 136E

In Exercise 5.42, the number of defects per yard in a certain fabric, Y, was known to have a Poisson distribution with parameter λ. The parameter λ was assumed to be a random variable with a density function given by

a Find the expected number of defects per yard by first finding the conditional expectation of Y for given λ.

b Find the variance of Y .

c Is it likely that Y exceeds 9?

Reference

The number of defects per yard Y for a certain fabric is known to have a Poisson distribution with parameter λ. However, λ itself is a random variable with probability density function given by

Find the unconditional probability function for Y .

Step-by-Step Solution:

Answer:

Step 1 of 3:

The number of defects ,follows a Poisson distribution with parameter

The parameter  was assumed to be a random variable with a density function given by

We need to find the conditional expectation of  for given

Let   and  denote random variables. Then

…….(1)

Where on the right-hand side the inside expectation is with respect to the conditional distribution of  given  and the outside expectation is with respect to the distribution of .

Let  be the number of defectives per yard and known to have a Poisson distribution with parameter

Hence using equation (1) we can write,

……….(2)

For a given  has a Poisson distribution.

We know the expected value of random variable , which follows a  Poisson distribution.

Hence we can write,

   

Thus equation (2) will become,

…….(3)

Also, we have given that  follows exponential with mean

Hence equation (3) will become,

Hence the conditional expectation of  for given  is

Step 2 of 3:

(b)

We need to find the variance of

Let   and  denote random variables. Then

…….(4)

Hence using equation (4) we can write,

…….(5)

We know the variance of random variable , which follows a  Poisson distribution.

Hence we can write,

   

Hence the variance of  is

Step 3 of 3

Chapter 5, Problem 136E is Solved
Textbook: Mathematical Statistics with Applications
Edition: 7
Author: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer
ISBN: 9780495110811

Mathematical Statistics with Applications was written by and is associated to the ISBN: 9780495110811. This full solution covers the following key subjects: Yard, given, defects, function, parameter. This expansive textbook survival guide covers 32 chapters, and 3350 solutions. This textbook survival guide was created for the textbook: Mathematical Statistics with Applications , edition: 7. Since the solution to 136E from 5 chapter was answered, more than 731 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 136E from chapter: 5 was answered by , our top Statistics solution expert on 07/18/17, 08:07AM. The answer to “In Exercise 5.42, the number of defects per yard in a certain fabric, Y, was known to have a Poisson distribution with parameter ?. The parameter ? was assumed to be a random variable with a density function given by a Find the expected number of defects per yard by first finding the conditional expectation of Y for given ?.b Find the variance of Y .c Is it likely that Y exceeds 9?ReferenceThe number of defects per yard Y for a certain fabric is known to have a Poisson distribution with parameter ?. However, ? itself is a random variable with probability density function given by Find the unconditional probability function for Y .” is broken down into a number of easy to follow steps, and 114 words.

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