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We have talked about the fact that the sample mean estimator X = X; is an unbiased estimator of the mean u for identically distributed X1, X2, ..., Xn: E[X] = variance estimator, on the other hand, is not an unbiased estimate of the true variance o²: for µ. The straightforward Vý = E1(X; – X)², we get that E[V½] = (1 – )o². Instead, the following bias-corrected sample variance estimator is unbiased: V = „4 E (X; – X)². This unbiased estimator is typically what is called the sample variance. Use the fact that E[V] = o² to show that E[V] = (1 – )0². Hint: The proof is short, it can be done in a few lines. (b) that Var[X] = 02, for iid variables with variance o?. We can similarly ask what the variance is We also discussed the variance of the sample mean estimator, and concluded

We have talked about the fact that the sample mean estimator X = X; is an unbiased estimator of the mean u for identically distributed X1, X2, ..., Xn: E[X] = variance estimator, on the other hand, is not an unbiased estimate of the true variance o²: for µ. The straightforward Vý = E1(X; – X)², we get that E[V½] = (1 – )o². Instead, the following bias-corrected sample variance estimator is unbiased: V = „4 E (X; – X)². This unbiased estimator is typically what is called the sample variance. Use the fact that E[V] = o² to show that E[V] = (1 – )0². Hint: The proof is short, it can be done in a few lines. (b) that Var[X] = 02, for iid variables with variance o?. We can similarly ask what the variance is We also discussed the variance of the sample mean estimator, and concluded

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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We have talked about the fact that the sample mean estimator X = X; is an unbiased estimator of the mean u for identically distributed X1, X2, ..., Xn: E[X] = variance estimator, on the other hand, is not an unbiased estimate of the true variance o²: for µ. The straightforward Vý = E1(X; – X)², we get that E[V½] = (1 – )o². Instead, the following bias-corrected sample variance estimator is unbiased: V = „4 E (X; – X)². This unbiased estimator is typically what is called the sample variance. Use the fact that E[V] = o² to show that E[V] = (1 – )0². Hint: The proof is short, it can be done in a few lines. (b) that Var[X] = 02, for iid variables with variance o?. We can similarly ask what the variance is We also discussed the variance of the sample mean estimator, and concluded
Transcribed Image Text:We have talked about the fact that the sample mean estimator X = X; is an unbiased estimator of the mean u for identically distributed X1, X2, ..., Xn: E[X] = variance estimator, on the other hand, is not an unbiased estimate of the true variance o²: for µ. The straightforward Vý = E1(X; – X)², we get that E[V½] = (1 – )o². Instead, the following bias-corrected sample variance estimator is unbiased: V = „4 E (X; – X)². This unbiased estimator is typically what is called the sample variance. Use the fact that E[V] = o² to show that E[V] = (1 – )0². Hint: The proof is short, it can be done in a few lines. (b) that Var[X] = 02, for iid variables with variance o?. We can similarly ask what the variance is We also discussed the variance of the sample mean estimator, and concluded
of the sample variance estimator. Deriving the formula is a bit more complex for general random variables, so let's assume the X; are zero-mean Gaussian. Note that for zero-mean Gaussian X;, we can use V =:E1 X}, which is unbiased (i.e., E[V] = o²). Then we know that the following is true (though we omit the derivation): Var[V] = 2(n-1),4. This variance enables us to use Chebyshev's inequality, to get a confidence estimate. Recall that Chebyshev's inequality states that for a random variable Y with known variance v, we know that Pr(|Y – E[Y]| < e) > 1 – v/e². After seeing 10 samples from a distribution, do you think you will have a tighter confidence estimate around the sample mean X or the sample variance V? n2
Transcribed Image Text:of the sample variance estimator. Deriving the formula is a bit more complex for general random variables, so let's assume the X; are zero-mean Gaussian. Note that for zero-mean Gaussian X;, we can use V =:E1 X}, which is unbiased (i.e., E[V] = o²). Then we know that the following is true (though we omit the derivation): Var[V] = 2(n-1),4. This variance enables us to use Chebyshev's inequality, to get a confidence estimate. Recall that Chebyshev's inequality states that for a random variable Y with known variance v, we know that Pr(|Y – E[Y]| < e) > 1 – v/e². After seeing 10 samples from a distribution, do you think you will have a tighter confidence estimate around the sample mean X or the sample variance V? n2
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