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# In Exercise 5.42, the number of defects per yard in a ISBN: 9780495110811 47

## Solution for problem 136E Chapter 5

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:

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

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

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