Solution Found!
(b) Find the amount of bias in the estimator.(c) What
Chapter 7, Problem 32E(choose chapter or problem)
(a) Show that \(\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} / n\) is a biased estimator of \(\sigma^{2}\). (b) Find the amount of bias in the estimator. (c) What happens to the bias as the sample size n increases?
Equation Transcription:
Text Transcription:
Sigma_i=1 ^n (X_1-X hat)^2/n
sigma^2
Questions & Answers
QUESTION:
(a) Show that \(\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} / n\) is a biased estimator of \(\sigma^{2}\). (b) Find the amount of bias in the estimator. (c) What happens to the bias as the sample size n increases?
Equation Transcription:
Text Transcription:
Sigma_i=1 ^n (X_1-X hat)^2/n
sigma^2
ANSWER:
Solution 32E
Step1 of 4:
From the given problem we have:
Here our goal is:
a). We need to show that is an unbiased estimator of
b). We need to find the amount of bias in the estimator.
c). We need to check What happens to the bias as the sample size n increases.
Step2 of 4:
a).
Consider,
Take expectation then we get,