(a) Show that X² is not an unbiased estimator for the area of the square plot p². [Hint: for any rv Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.] (b) For what value of k is the estimator X² – kS² unbiased for µ²?
(a) Show that X² is not an unbiased estimator for the area of the square plot p². [Hint: for any rv Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.] (b) For what value of k is the estimator X² – kS² unbiased for µ²?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Topic Video
Question
![(Sec. 6.1) Using a long rod that has length \( \mu \) (unknown), you are going to lay out a square plot in which the length of each side is \( \mu \). Thus the area of the plot will be \( \mu^2 \). However, because you do not know the value of \( \mu \), you decide to make \( n \) independent measurements \( X_1, \ldots, X_n \) of the length. Assume that each \( X_i \) has mean \( \mu \) and variance \( \sigma^2 \).
(a) Show that \( \bar{X}^2 \) is not an unbiased estimator for the area of the square plot \( \mu^2 \). [Hint: for any random variable \( Y \), \( E[Y^2] = V[Y] + E[Y]^2 \). Apply this for \( Y = \bar{X} \).]
(b) For what value of \( k \) is the estimator \( \bar{X}^2 - kS^2 \) unbiased for \( \mu^2 \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4f8485b-d935-478d-91e7-b806502d9702%2F956ca2fa-b50b-4caa-9b5e-39554cd27dd5%2Fgkqcd2_processed.png&w=3840&q=75)
Transcribed Image Text:(Sec. 6.1) Using a long rod that has length \( \mu \) (unknown), you are going to lay out a square plot in which the length of each side is \( \mu \). Thus the area of the plot will be \( \mu^2 \). However, because you do not know the value of \( \mu \), you decide to make \( n \) independent measurements \( X_1, \ldots, X_n \) of the length. Assume that each \( X_i \) has mean \( \mu \) and variance \( \sigma^2 \).
(a) Show that \( \bar{X}^2 \) is not an unbiased estimator for the area of the square plot \( \mu^2 \). [Hint: for any random variable \( Y \), \( E[Y^2] = V[Y] + E[Y]^2 \). Apply this for \( Y = \bar{X} \).]
(b) For what value of \( k \) is the estimator \( \bar{X}^2 - kS^2 \) unbiased for \( \mu^2 \)?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.Recommended textbooks for you
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
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