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Find the distribution of W = X2 when(a) X is N(0, 4),(b) X

Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman ISBN: 9780321923271 41

Solution for problem 9E Chapter 3.3

Probability and Statistical Inference | 9th Edition

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Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

Probability and Statistical Inference | 9th Edition

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Problem 9E

Find the distribution of W = X2 when

(a) X is N(0, 4),

(b) X is N(0, σ2).

Step-by-Step Solution:

Answer :

Step 1 of 3 :

A random variable X is  normally distributed with mean  and standard deviation  .

The probability density function is

f(x) =

The claim is to find the distribution of  w =  when.

X ~ N(0, 4)X ~ N(0, )

         In general,    ~ with one degrees of freedom.

Where, v =

The probability density function of v is

P( V  v) =  dy   ,       0 < v <

                 =  dy

Step 2 of 3 :

Here, X follows N(0,4)

          w = , v =  it follows with one degrees of freedom.

Let w =

          = 4v

Then, P(W  w) = P(4v  w)

                             = P(v  )

                            =  dy   -  (1)

Let, z = 4y

     

      = dy

Substitute in equation 1

                         =     

                        =     

Which follows gamma distribution with = ½  and = 8

Therefore,

P(W  w) =   be the distribution function of w =  when

X ~ N(0, 4).

Step 3 of 3

Chapter 3.3, Problem 9E is Solved
Textbook: Probability and Statistical Inference
Edition: 9
Author: Robert V. Hogg, Elliot Tanis, Dale Zimmerman
ISBN: 9780321923271

The answer to “Find the distribution of W = X2 when(a) X is N(0, 4),(b) X is N(0, ?2).” is broken down into a number of easy to follow steps, and 16 words. This textbook survival guide was created for the textbook: Probability and Statistical Inference , edition: 9. Since the solution to 9E from 3.3 chapter was answered, more than 301 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 9E from chapter: 3.3 was answered by , our top Statistics solution expert on 07/05/17, 04:50AM. This full solution covers the following key subjects: distribution, Find. This expansive textbook survival guide covers 59 chapters, and 1476 solutions. Probability and Statistical Inference was written by and is associated to the ISBN: 9780321923271.

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