MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
expand_more
expand_more
format_list_bulleted
Related questions
Question
Please show all work when answering this question.
The question is shown at the top and is followed by a hint to help with the solution.
![The image contains a mathematical explanation related to estimating the unknown parameter \(\lambda\) of a Poisson distribution, often used to model the number of wrong phone connections.
**Content Details:**
1. **Introduction**
- The text discusses modeling the number of wrong phone connections using a Poisson distribution.
- The goal is to estimate the unknown parameter \(\lambda\), representing the mean number of wrong connections.
2. **Tasks**
- (i) Provide the likelihood function of a sample.
- (ii) Find the maximum likelihood estimator (MLE) of \(\lambda\).
3. **Hint for Step (i)**
- If \(f(x|\lambda) = e^{-\lambda}\lambda^x/x!\), then use this to derive the likelihood function.
- The likelihood function is given as:
\[
f_n(\mathbf{x}_n|\lambda) = e^{-n\lambda}\lambda^\sigma/\prod_{j=1}^{n}x_j! \quad (\sigma = x_1 + \ldots + x_n)
\]
4. **Step (ii)**
- Define the log-likelihood function:
\[
L(\lambda) = \ln f_n(\mathbf{x}_n|\lambda)
\]
- Find its derivative \(L'(\lambda)\) and show that:
\[
L'(\lambda) = 0 \quad \text{iff} \quad \lambda = \sigma/n = \overline{x}_n
\]
5. **Critical Points and Behavior**
- \( \sigma/n \) is a critical point of \( L(\lambda) \).
- Rewrite \( L'(\lambda) \) as:
\[
L'(\lambda) = \frac{1}{\lambda_n}(\overline{x}_n - \lambda)
\]
- Analysis of \( L' \):
- Positive for \( \lambda < \overline{x}_n \)
- Zero at \( \lambda = \overline{x}_n \)
- Negative for \( \lambda > \overline{x}_n \)
6. **Conclusion**
- It’s proven that \(\overline{x}_n\) is a local and global maximum point.
- Therefore, \(\hat{\lambda} = \overline{x}_n\) is](https://content.bartleby.com/qna-images/question/db82eee3-c597-43b7-b98f-cffc18fbca72/0081d0a0-9e87-46f9-b6d8-81d947b4ed6e/jqbr5g_processed.png)
Transcribed Image Text:The image contains a mathematical explanation related to estimating the unknown parameter \(\lambda\) of a Poisson distribution, often used to model the number of wrong phone connections.
**Content Details:**
1. **Introduction**
- The text discusses modeling the number of wrong phone connections using a Poisson distribution.
- The goal is to estimate the unknown parameter \(\lambda\), representing the mean number of wrong connections.
2. **Tasks**
- (i) Provide the likelihood function of a sample.
- (ii) Find the maximum likelihood estimator (MLE) of \(\lambda\).
3. **Hint for Step (i)**
- If \(f(x|\lambda) = e^{-\lambda}\lambda^x/x!\), then use this to derive the likelihood function.
- The likelihood function is given as:
\[
f_n(\mathbf{x}_n|\lambda) = e^{-n\lambda}\lambda^\sigma/\prod_{j=1}^{n}x_j! \quad (\sigma = x_1 + \ldots + x_n)
\]
4. **Step (ii)**
- Define the log-likelihood function:
\[
L(\lambda) = \ln f_n(\mathbf{x}_n|\lambda)
\]
- Find its derivative \(L'(\lambda)\) and show that:
\[
L'(\lambda) = 0 \quad \text{iff} \quad \lambda = \sigma/n = \overline{x}_n
\]
5. **Critical Points and Behavior**
- \( \sigma/n \) is a critical point of \( L(\lambda) \).
- Rewrite \( L'(\lambda) \) as:
\[
L'(\lambda) = \frac{1}{\lambda_n}(\overline{x}_n - \lambda)
\]
- Analysis of \( L' \):
- Positive for \( \lambda < \overline{x}_n \)
- Zero at \( \lambda = \overline{x}_n \)
- Negative for \( \lambda > \overline{x}_n \)
6. **Conclusion**
- It’s proven that \(\overline{x}_n\) is a local and global maximum point.
- Therefore, \(\hat{\lambda} = \overline{x}_n\) is
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 4 steps
Knowledge Booster
Similar questions
Recommended textbooks for you
- MATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
Probability and Statistics for Engineering and th...StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage Learning
Statistics for The Behavioral Sciences (MindTap C...StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage Learning 
Elementary Statistics: Picturing the World (7th E...StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSON
The Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. Freeman
Introduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman
MATLAB: An Introduction with Applications
Statistics
ISBN:9781119256830
Author:Amos Gilat
Publisher:John Wiley & Sons Inc
Probability and Statistics for Engineering and th...
Statistics
ISBN:9781305251809
Author:Jay L. Devore
Publisher:Cengage Learning
Statistics for The Behavioral Sciences (MindTap C...
Statistics
ISBN:9781305504912
Author:Frederick J Gravetter, Larry B. Wallnau
Publisher:Cengage Learning
Elementary Statistics: Picturing the World (7th E...
Statistics
ISBN:9780134683416
Author:Ron Larson, Betsy Farber
Publisher:PEARSON
The Basic Practice of Statistics
Statistics
ISBN:9781319042578
Author:David S. Moore, William I. Notz, Michael A. Fligner
Publisher:W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:9781319013387
Author:David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:W. H. Freeman