mean_and_variance_of_the_sample_mean
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mean_and_variance_of_the_sample_mean [2020/04/22 14:51] – [Mean and variance of sample mean] hkimscil | mean_and_variance_of_the_sample_mean [2020/12/05 18:42] (current) – [Variance of the sample mean] hkimscil | ||
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</ | </ | ||
- | ====== Mean of Sample Mean ====== | + | ====== Mean of the sample mean ====== |
평균이 (mean) $\mu$ 이고, 분산이 (variance) $\sigma^{2}$ 인 모집단에서 (population) 독립적으로 추출되어 관촬되는 $X_{1}, X_{2}, . . . , X_{n}$ 이 있다고 하자. 이 샘플링은 아래와 같이 도식화되어 생각될 수 있다. | 평균이 (mean) $\mu$ 이고, 분산이 (variance) $\sigma^{2}$ 인 모집단에서 (population) 독립적으로 추출되어 관촬되는 $X_{1}, X_{2}, . . . , X_{n}$ 이 있다고 하자. 이 샘플링은 아래와 같이 도식화되어 생각될 수 있다. | ||
\begin{eqnarray*} | \begin{eqnarray*} | ||
Line 25: | Line 25: | ||
\begin{eqnarray*} | \begin{eqnarray*} | ||
- | E[X_{i}] & = & \mu \\ | + | E\left[X_{i}\right] & = & \mu \\ |
- | Var[X_{i}] & = & \sigma^{2} | + | Var\left[X_{i}\right] & = & \sigma^{2} |
\end{eqnarray*} | \end{eqnarray*} | ||
한편, $\overline{X}$ 는 | 한편, $\overline{X}$ 는 | ||
- | \begin{eqnarray*} | + | \begin{align*} |
- | \overline{X} = \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \\ | + | \overline{X} |
- | \end{eqnarray*} | + | \end{align*} |
+ | 이고 | ||
- | \begin{eqnarray*} | + | \begin{align*} |
- | E[\overline{X}] & = & E \left[ \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \right] \\ | + | E\left[\overline{X}\right] & = E \left[ \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \right] \\ |
- | & = & \left( \frac{1}{n} \right) E \left[ X_{1} + X_{2} + . . . + X_{n} \right] \\ | + | & = \left( \frac{1}{n} \right) E \left[ X_{1} + X_{2} + . . . + X_{n} \right] \\ |
- | & = & \left( \frac{1}{n} \right) \left(E \left[ X_{1} + X_{2} + . . . + X_{n} \right] \right)\\ | + | & = \left( \frac{1}{n} \right) \left(E \left[X_{1} + X_{2} + . . . + X_{n} \right] \right) \\ |
- | & = & \left( \frac{1}{n} \right) \left(E[X_{1}] + E[X_{2}] + . . . + E[X_{n}]\right)\\ | + | & = \left( \frac{1}{n} \right) \left(E[X_{1}] + E[X_{2}] + . . . + E[X_{n}]\right) \\ |
- | & = & \left( \frac{1}{n} \right) \left(\mu + \mu + . . . + \mu\right)\\ | + | & = \left( \frac{1}{n} \right) \left(\mu + \mu + . . . + \mu\right)\\ |
- | & = & \left( \frac{1}{n} \right) (n \mu)\\ | + | & = \left( \frac{1}{n} \right) (n \mu)\\ |
- | & = & \mu | + | & = \mu \\ \\ \\ |
- | \end{eqnarray*} | + | E\left[\overline{X}\right] & = \mu_{\overline{X}} = \mu \\ |
- | \begin{eqnarray*} | + | \end{align*} |
- | E[\overline{X}] & = & \mu \\ | + | |
- | \mu_{\overline{X}} | + | |
- | \end{eqnarray*} | + | |
- | ====== Variance of sample mean ====== | ||
- | \begin{eqnarray*} | ||
- | Var[\overline{X}] & = & Var \left[ \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \right] \\ | ||
- | & = & (\frac{1}{n})^2 Var \left[ X_{1} + X_{2} + . . . + X_{n} \right] \\ | ||
- | & = & (\frac{1}{n})^2 (Var[X_{1} + X_{2} + . . . + X_{n}]) \\ | ||
- | & = & (\frac{1}{n})^2 (Var[X_{1}] + Var[X_{2}] + . . . + Var[X_{n}])\\ | ||
- | & = & (\frac{1}{n})^2 (\sigma^2 + \sigma^2 + . . . + \sigma^2) \\ | ||
- | & = & \frac{1}{n^2} n \sigma^2 \\ | ||
- | & = & \frac{\sigma^2}{n} | ||
- | \end{eqnarray*} | ||
- | \begin{eqnarray*} | + | ====== Variance of the sample mean ====== |
- | Var[\overline{X}] & = & \frac{\sigma^2}{n} \\ | + | |
- | \sigma_{\overline{X}}^{2} & = & \frac{\sigma^2}{n} \\ | + | \begin{align*} |
- | \sigma_{\overline{X}} | + | Var\left[\overline{X}\right] & = Var \left[ \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \right] \\ |
- | \end{eqnarray*} | + | & = \left(\frac{1}{n}\right)^2 Var \left[ X_{1} + X_{2} + . . . + X_{n} \right] \\ |
+ | & = \left(\frac{1}{n}\right)^2 \left(Var[X_{1} + X_{2} + . . . + X_{n}]\right) \\ | ||
+ | & = \left(\frac{1}{n}\right)^2 \left(Var[X_{1}] + Var[X_{2}] + . . . + Var[X_{n}]\right)\\ | ||
+ | & = \left(\frac{1}{n}\right)^2 \left(\sigma^2 + \sigma^2 + . . . + \sigma^2\right) \\ | ||
+ | & = \frac{1}{n^2} n \sigma^2 \\ | ||
+ | & = \frac{\sigma^2}{n} | ||
+ | \\ | ||
+ | \\ | ||
+ | Var\left[\overline{X}\right] | ||
+ | \sigma_{\overline{X}}^{2} & = \frac{\sigma^2}{n} \\ | ||
+ | \sigma_{\overline{X}} | ||
+ | \end{align*} | ||
+ |
mean_and_variance_of_the_sample_mean.txt · Last modified: 2020/12/05 18:42 by hkimscil