Let X1,..., Xn be a random sample from a N(0, ?2)

Problem 9E Chapter 4.9

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition

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Problem 9E

Let X1,..., Xn be a random sample from a N(0, σ2) population. Find the MLE of σ.

Step-by-Step Solution:

Answer:

Step 1 of 1:

Let X follows Normal distribution with Probability density function

f(x) = $\frac{1}{\sqrt{2\Pi{}}}$ ${e}^{-\ \frac{\Sigma{}({x}_{i}-\ \mu{}{)}^{2}}{2}}$

The claim is to find the Maximum likelihood function (MLE) of $\sigma{}$ .

The likelihood function is

L(x: $\sigma{}$ ) = $\sum\limits_{i\ =1}^{n}$ ( $\frac{1}{\sigma{}(\sqrt{2\Pi{}{)}^{\ }}}$ ${e}^{-\ \Sigma{}\frac{({x}_{i}){\ }^{2}}{2{\sigma{}}^{2}}}$ )

Take log on both sides

Log L(x: $\sigma{}$ ) = Log( $(2\Pi{}{)}^{-\frac{n}{2}}$ ${e}^{-\ \Sigma{}\frac{({x}_{i}{)}^{2}}{2{\sigma{}}^{2}}}$ )

= - $\frac{n}{2}$ log (2 $\Pi{}$ ${)}^{\ }$ - n log( $\sigma{}$ ) $-\ \Sigma{}\frac{({x}_{i}){\ }^{2}}{2{\sigma{}}^{2}}$

Differentiate with respect to $\sigma{}$ and equate it to zero.

$\frac{d}{d\sigma{}}$ (- $\frac{n}{2}$ log (2 $\Pi{}$ ${)}^{\ }$ - n log( $\sigma{}$ )...

Step 2 of 3

Chapter 4.9, Problem 9E is Solved

Step 3 of 3

Textbook: Statistics for Engineers and Scientists

Edition: 4th

Author: William Navidi

ISBN: 9780073401331

The answer to “Let X1,..., Xn be a random sample from a N(0, ?2) population. Find the MLE of ?.” is broken down into a number of easy to follow steps, and 17 words. This full solution covers the following key subjects: Find, let, mle, population, random. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4th. Since the solution to 9E from 4.9 chapter was answered, more than 223 students have viewed the full step-by-step answer. Statistics for Engineers and Scientists was written by Patricia and is associated to the ISBN: 9780073401331. The full step-by-step solution to problem: 9E from chapter: 4.9 was answered by Patricia, our top Statistics solution expert on 06/28/17, 11:15AM.