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(ii) If $x_{i}$ follows Bernoulli $(p)$, show that $$ \frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}=\bar{x}(1- \bar{x}) $$ so that the unbiased estimator of $\operatorname{Var}(\bar{x})$ is $$ \frac{1}{n} s^{2}=\frac{1}{n}\left(\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right)=\frac{\bar{x}(1- \bar{x})}{n-1} . $$ SP. JG.079

(ii) If $x_{i}$ follows Bernoulli $(p)$, show that $$ \frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}=\bar{x}(1- \bar{x}) $$ so that the unbiased estimator of $\operatorname{Var}(\bar{x})$ is $$ \frac{1}{n} s^{2}=\frac{1}{n}\left(\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right)=\frac{\bar{x}(1- \bar{x})}{n-1} . $$ SP. JG.079

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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(ii) If $x_{i}$ follows Bernoulli $(p)$, show that $$ \frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}=\bar{x}(1- \bar{x}) $$ so that the unbiased estimator of $\operatorname{Var}(\bar{x})$ is $$ \frac{1}{n} s^{2}=\frac{1}{n}\left(\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right)=\frac{\bar{x}(1- \bar{x})}{n-1} . $$ SP. JG. 079
Transcribed Image Text:(ii) If $x_{i}$ follows Bernoulli $(p)$, show that $$ \frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}=\bar{x}(1- \bar{x}) $$ so that the unbiased estimator of $\operatorname{Var}(\bar{x})$ is $$ \frac{1}{n} s^{2}=\frac{1}{n}\left(\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right)=\frac{\bar{x}(1- \bar{x})}{n-1} . $$ SP. JG. 079
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