표본분산과 자유도(n - 1)

표본분산과 자유도(n - 1)

1. 평균
   \( \mu = \frac{\sum_{k=1}^{n}}{n} \)

2. 분산
   \( \sigma^2 = \frac{\sum_{k=1}^{n} (X_k -\mu)^2}{n} \)

3. 표본평균의 평균
   \( \begin{align} \bar{X}_\bar{X} & = E(\bar{X}) \\ & = E(\sum_{k=1}^{n} \frac{X_k}{n}) \\ & = E[ \frac{X_1 + X_2 + \ldots + X_n}{n}] \\ & = E(\frac{1}{n}X_1) + E(\frac{1}{n}X_2) + \ldots + E(\frac{1}{n}X_n) \\ & = \frac{1}{n}\mu_x + \frac{1}{n}\mu_x + \ldots + \frac{1}{n}\mu_x \\ & = \mu_x \end{align} \)

4. 표본평균의 분산
   \( \begin{align} \sigma_\bar{X}^2 & = Var(\bar{X}) \\ & = Var(\frac{X_1 + X_2 + \ldots + X_n}{n}) \\ & = Var(\frac{1}{n}X_1) + Var(\frac{1}{n}X_2) + \ldots + Var(\frac{1}{n}X_n) \\ & = (\frac{1}{n})^2Var(X_1) + (\frac{1}{n})^2Var(X_2) + \ldots + (\frac{1}{n})^2Var(X_n) \\ & = n (\frac{1}{n})^2 Var(X) \\ & = \frac{Var(X)}{n} \\ & = \frac{\sigma^2}{n} \end{align} \)

5. 표본평균의 표준편차
   \( \begin{align} \sigma_\bar{X} & = \frac{\sqrt{Var(\bar{X})}}{\sqrt{n}} \\ & = \frac{\sigma}{\sqrt{n}} \end{align} \)

6. 표본분산
   \( \begin{align} s^2 & = E[(X-\bar{X})^2] \\ & = E[{(X - \mu) + (\mu - \bar{X})}^2] \\ & = \frac{1}{n} \sum_{k=1}^{n}(X_k - \bar{X})^2 \\ & = \frac{1}{n} \sum_{k=1}^{n} [(X_k - \mu) + (\mu - \bar{X})]^2 \\ & = \frac{1}{n} [ \sum_{k=1}^{n}(X_k - \mu)^2 + 2(\mu - \bar{X})n(\bar{X} - \mu) + n(\mu - \bar{X})^2 ] \\ & = \frac{1}{n} [ \sum_{k=1}^{n}(X_k - \mu)^2 - 2n(\mu - \bar{X})^2 + n(\mu - \bar{X})^2 ] \\ & = \frac{1}{n} [ \sum_{k=1}^{n}(X_k - \mu)^2 - n(\bar{X} - \mu)^2 ] \\ & = \sigma^2 - \frac{\sigma^2}{n} \\ & = \frac{(n - 1)}{n} \sigma^2 \end{align} \)

즉,
\( \frac{\sigma^2}{n} = \frac{s^2}{n - 1} \)