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Let X1,..., Xn be a random sample from a N(µ, ?2)

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 10E Chapter 4.9

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 10E

Problem 10E

Let X1,..., Xn be a random sample from a N(µ, σ2) population. Find the MLEs of µ and of σ. (Hint: The likelihood function is a function of two parameters, µ and σ. Compute partial derivatives with respect to µ and σ and set them equal to 0 to find the values  and  that maximize the likelihood function.)

Step-by-Step Solution:

Solution :

Step 1 of 1:

Let be a random sample from a normal distribution with mean and .

The density for X is

f(x)=

Our goal is:

We need to find the maximum likelihood estimator of mean and variance .

Now we have to find the maximum likelihood estimator of mean and variance .

Here the likelihood function for the sample is

L(,)=

=

=

We are applying the logarithm likelihood function is

Ln L(,)=-nln.

We compute the first partial derivatives with respect to mean and variance and we set equal to 0.

Then,

= 

Here

=

= 0

Then,

=

=

Here                                 =0

= 0

 

When and Then the second partial derivatives is

=and

=

=

=

Here Xand also given random sample with size n.

Then the the maximum likelihood estimator of mean and variance are  and  respectively.


Step 2 of 1

Chapter 4.9, Problem 10E is Solved
Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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