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Use the result of Exercise 2.5-5 to find the mean and

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

Solution for problem 5E Chapter 2.5

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 5E

Let the moment-generating function M(t) of X exist for \(-h<t<h\). Consider the function \(R(t)=\ln M(t)\). The first two derivatives of R(t) are, respectively,

\(R^{\prime}(t)=\frac{M^{\prime}(t)}{M(t)} \text { and } R^{\prime \prime}(t)=\frac{M(t) M^{\prime \prime}(t)-\left[M^{\prime}(t)\right]^{2}}{[M(t)]^{2}}\)

Setting , show that

(a) .

(b) .

Equation Transcription:

 

.

Text Transcription:

-h < t < h

R(t) = ln M(t)

R prime prime (t) = M(t)M prime prime (t) - [M prime (t)]^2/[M(t)]^2

Nu = R prime (0)

Sigma^2 = R prime prime (0)

Step-by-Step Solution:

Answer:

Step 1 of 6:

                               From exercise 2.5 - 5.

            

and  

 Put t = 0.

       

           Then,

                   

                          =

                              =    

 

                              = .

                                                                 Therefore,  =

  

                  =

               =

                                                              =

                                              Therefore,  

   =  and

          is the moment generating function.


Step 2 of 5

Chapter 2.5, Problem 5E is Solved
Step 3 of 5

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

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Use the result of Exercise 2.5-5 to find the mean and