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Find the moment-generating function for the gamma

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

Solution for problem 7E 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 7E

Problem 7E

Find the moment-generating function for the gamma distribution with parameters α and θ.

Hint: In the integral representing E(etX ), change variables by letting y = (1 − θt)x/θ, where 1 − θt > 0.

Step-by-Step Solution:
Step 1 of 3

Solution 7E

Step1 of 2:

We have A random variable X which follows gamma distribution with parameter

We need to find the moment generating function(mgf) of  gamma distribution.

Step2 of 2:

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

That is X G()

The probability mass function of binomial distribution is given below

P(X) = , 

Where,

X = random variable

 = parameter

= parameter

e = a constant it is approximately 2.7182.

Now,

The moment generating function is given by

          =

       =

Let  = y

If (1-) > 0

=

       =

Where,

Substitute  value in above equation we get

          =

Simplifying above equation we get

         =

Therefore,the moment generating function of X is .

Conclusion:

The moment generating function of X is .

 

 

Step 2 of 3

Chapter 3.2, Problem 7E 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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Find the moment-generating function for the gamma