Sample Variance

Definition

Sample Variance $s^2 = \frac{\sum(X_{i}-\bar{X})}{n-1}$

 

n: size of sample

df: n-1

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Proof

1. $E(S^2)=\sigma^2$

$E(S^2)$ 가 $\sigma^2$의 점추정치가 될 수 있다.

 

proof)

$E(S^2)$

$=E(\frac{1}{n-1}\sum(X_{i}-bar{X})^2)$

$=\frac{1}{n-1}E(\sum(X_{i}-\mu+\mu-\bar{X})^2$

$=\frac{1}{n-1}E(\sum((X_{i}-\mu)^2+(\mu-\bar{X})^2+2(X_{i}-\mu)(\mu-\bar{X})))$

$=\frac{1}{n-1}E(\sum((X_{i}-\mu)^2)+n(\mu-\bar{X})-2n(\mu-\bar{X})^2)$

$=\frac{1}{n-1}E(\sum((X_{i}-\mu)^2))+\frac{1}{n-1}E(-n(\mu-\bar{X})^2)$

$=\frac{1}{n-1}E(\sum((X_{i}-\mu)^2))-\frac{n}{n-1}E((\mu-\bar{X})^2)$

$=\frac{1}{n-1}n\sigma^2-\frac{n}{n-1}\frac{\sigma^2}{n}$

$=\frac{n}{n-1}\sigma^2-\frac{\sigma^2}{n-1}$

$=(\frac{n-1}{n-1})\sigma^2$

$=\sigma^2$

 

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lemma

1) $E(\sum(X_{i}-\mu)^2)=n\sigma^2$

$var(X)$

$=E(\frac{1}{n}\sum(X_{i}-\mu)^2)$

$=\frac{1}{n}E(\sum(X_{i}-\mu)^2)$

$=\sigma^2$

2) $E(\sum(\mu-\bar{x})^2)=E(n(\mu-\bar{X})^2)$

3) $E(\sum(2(X_{i}-\mu)(\mu-\bar{X}))$

$E(\sum(2(X_{i}-\mu)(\mu-\bar{X})))$

$=E(2(\mu-\bar{X})\sum((X_{i}-\mu)))$

$=E(2(\mu-\bar{X})(n\bar{X}-n\mu))$

$=E(2n(\mu-\bar{X})(\bar{X}-\mu))$

$=E(-2n(\mu-\bar{X})^2)$

4) $var(\bar{X})=E((\bar{X}-\mu)^2)=\frac{\sigma^2}{n}$