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
![### Educational Website Content
#### Topic: Maximum Likelihood Estimation and Vibratory Stress Distribution
---
**Problem Statement:**
Let \( X_1, X_2, \ldots, X_n \) represent a random sample from a Rayleigh distribution with the following probability density function (pdf):
\[ f(x; \theta) = \frac{x}{\theta} e^{-x^2/(2\theta)} \quad \text{ for } \quad x > 0 \]
**Tasks:**
---
**(a) Determine the maximum likelihood estimator of \( \theta \).**
Options:
- \(\circ \quad \bar{X} / n \)
- \(\circ \quad (\Sigma X_i^2) / 2n \)
- \(\circ \quad (\Sigma X_i^2) / n \)
- \(\circ \quad \bar{X} / 2n \)
- \(\circ \quad (\Sigma X_i^2) / 2n \)
**(b) Calculate the estimate from the following \( n = 10 \) observations on vibratory stress of a turbine blade under specified conditions. (Round your answer to three decimal places.)**
Observations:
- 12.38, 9.01, 11.00, 5.04, 6.31
- 7.56, 8.68, 8.25, 16.38, 11.86
(Enter the calculated estimate in the provided box.)
**(c) Determine the maximum likelihood estimate (MLE) of the median of the vibratory stress distribution.**
Hint: First, express the median in terms of \( \theta \).
Options:
- \(\circ \quad 1.3863 \hat{\theta} \)
- \(\circ \quad (1.3863 \hat{\theta})^3 \)
- \(\circ \quad 3 \sqrt{1.3863 \hat{\theta}} \)
- \(\circ \quad (1.3863 \hat{\theta})^2 \)
- \(\circ \quad 1.3863 \hat{\theta}^2 \)
---
This problem set guides you through the process of deriving the maximum likelihood estimator for the Rayleigh distribution parameter, applying the estimator to given data, and determining the MLE of the median of the distribution.](https://content.bartleby.com/qna-images/question/fb989860-4d1c-4fb5-9e6b-42a4528dce9c/71c75b1a-6041-400d-946f-af020ffd75db/y1ozwm_processed.jpeg)
Transcribed Image Text:### Educational Website Content
#### Topic: Maximum Likelihood Estimation and Vibratory Stress Distribution
---
**Problem Statement:**
Let \( X_1, X_2, \ldots, X_n \) represent a random sample from a Rayleigh distribution with the following probability density function (pdf):
\[ f(x; \theta) = \frac{x}{\theta} e^{-x^2/(2\theta)} \quad \text{ for } \quad x > 0 \]
**Tasks:**
---
**(a) Determine the maximum likelihood estimator of \( \theta \).**
Options:
- \(\circ \quad \bar{X} / n \)
- \(\circ \quad (\Sigma X_i^2) / 2n \)
- \(\circ \quad (\Sigma X_i^2) / n \)
- \(\circ \quad \bar{X} / 2n \)
- \(\circ \quad (\Sigma X_i^2) / 2n \)
**(b) Calculate the estimate from the following \( n = 10 \) observations on vibratory stress of a turbine blade under specified conditions. (Round your answer to three decimal places.)**
Observations:
- 12.38, 9.01, 11.00, 5.04, 6.31
- 7.56, 8.68, 8.25, 16.38, 11.86
(Enter the calculated estimate in the provided box.)
**(c) Determine the maximum likelihood estimate (MLE) of the median of the vibratory stress distribution.**
Hint: First, express the median in terms of \( \theta \).
Options:
- \(\circ \quad 1.3863 \hat{\theta} \)
- \(\circ \quad (1.3863 \hat{\theta})^3 \)
- \(\circ \quad 3 \sqrt{1.3863 \hat{\theta}} \)
- \(\circ \quad (1.3863 \hat{\theta})^2 \)
- \(\circ \quad 1.3863 \hat{\theta}^2 \)
---
This problem set guides you through the process of deriving the maximum likelihood estimator for the Rayleigh distribution parameter, applying the estimator to given data, and determining the MLE of the median of the distribution.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 3 steps with 6 images
Knowledge Booster
Similar questions
- If X ∼ Gamma(2, 5), find E(X^2 ) andE(X^4 ) Do I simply use the pdf for gamma square it, then find the expected value of that?arrow_forward6) (13 points) The random variable X has the following probability density function: x 1x(x)=2 0, for 0arrow_forwardPlease help me with this one. Thank you tutor!arrow_forwardFind the mean and standard deviation for each uniform continuous model. (Round "Mean" answers to 1 decimal place and "Standard deviation" answers to 4 decimal places.) Mean Standard Deviation a. U(2, 12) b. U(100, 250) C. U(2, 99) www Next > < Prev 2 of 8 acer 31arrow_forwardSecond part of c, as well as parts d and earrow_forwardPvgarrow_forwardPlot the gamma distribution by fixing the shape parameter *k* = 3 and setting the scale parameter = 0.5, 1, 2, 3, 4, 5. What is the effect of increasing the scale parameter?arrow_forwardjust question 2 please thank you (calculate the mean and the variance of the normal distribution from its moment generating function - refer to picture) thank you so mucharrow_forwardQuestion 7 b Question submitted. > Next question You can retry this question below Let X be a continuous random variable with a probability density function f. Let the x-value a be such that P(X ≤ a) = 0.5. Let the x-value c be such that f(c) > f(x) for all x. = 100 X- f(x) dx Mean = Median = a Mode = СУ ▼ bv > Submit Question Q Search 0 LEA [ W 8.arrow_forwardQuestion * Given X be the life time of a bulb having a probability density function f(x) =-e for x>0 and 0 elsewhere. The mean life time of the bulbs is equal to: %3D O 1/2 O None of these O 1/4 Question* Let X and Y be two independent continuous random variables with marginal distributionarrow_forwardand variance of X if the probability mass function is f(x) = ; ;)" , Find the moment generating function, mean x = 1, 2, 3, 4, ...arrow_forwardDon’t know how to find these valuesarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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