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There are times when a shifted exponential model is

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

Solution for problem 5E 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 5E

Problem 5E

There are times when a shifted exponential model is appropriate. That is, let the pdf of X be

(a) Define the cdf of X.

(b) Calculate the mean and variance of X.

Step-by-Step Solution:
Step 1 of 3

Solution 5E

Step1 of 3:

We have A Exponential random variable X with pdf

 

We need to find,

(a) Define the cdf of X.

(b) Calculate the mean and variance of X.

Step2 of 3:

Let “X” be random variable which follows Exponential distribution with parameters .

That is X Exp()

The probability mass function of binomial distribution is given below

Where,

X = random variable

 = parameter

 = parameter

e = a constant it is approximately 2.7182.

a).

Consider,

To get a cdf of X we need to integrate above function with respect to x.

1).If X < then,

   

               =

               = 0.

2).If X then,

   

               =

          =  

          = [

          =

          =

          =

Hence, the cdf is given by


Step3 of 3: 

b).

Consider,

To get mean and variance of X we need to integrate above function with respect to x.

 

            =

            = []

        =

           =

           = /

           =

Hence,  .

 

              =

              =    

              = -[0 - ]+

              = + E(X)

              = + []      

              = +      

Hence, +

Therefore, the variance of X is given by:

V(X) = -

         = +  -

         =

         =

Hence, V(X) = .


Conclusion:

a).The cdf is given by

     

b).Variance

Step 2 of 3

Chapter 3.2, Problem 5E is Solved
Step 3 of 3

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

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There are times when a shifted exponential model is