Let the distribution of X be ?2(r).(a) Find the point at

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

Problem 15E Chapter 3.2

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 15E

Let the distribution of X be χ2(r).

(a) Find the point at which the pdf of X attains its maximum when r ≥ 2. This is the mode of a χ2(r) distribution.

(b) Find the points of inflection for the pdf of X when r ≥ 4.

(c) Use the results of parts (a) and (b) to sketch the pdf of X when r = 4 and when r = 10.

Step-by-Step Solution:

Answer:

Step 1 of 4:

              Let the distribution of ‘X’ be

           The probability density function of distribution with ‘r’ degrees of freedom is

              

     Step 2 of 4:

      a). Now we have to find the derivative of the pdf of the distribution.

                 =

                            =  

       

                           = ((

                               f(x)  = 

            Equating the derivative to zero to find the extreme point, that is = 0.  

                               = 0

                   

                      

                      = 0

                  Therefore, x = r - 2.

                     We know that f(0) = 0 , and () can ever be zero at x > 0, so the critical point is x = r - 2. Here the pdf attains is maximum for x = r - 2, where ‘r’ is the degrees of freedom.

               Therefore, the mode of distribution is x = r - 2.

 

Step 3 of 4

Chapter 3.2, Problem 15E is Solved
Step 4 of 4

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

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