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--- mean_and_variance_of_the_sample_mean [2020/04/12 08:07] – created hkimscil
+++ mean_and_variance_of_the_sample_mean [2020/12/05 18:42] (current) – [Variance of the sample mean] hkimscil
@@ Line 1: / Line 1: @@
 ====== Mean and variance of sample mean ======
+전제: Expected value (기대값)와 Variance (분산)의 연산에 과한 법칙으로는 (([[:statistical review]])) 참조.
+<WRAP box 450px>
+X,Y are Independent variables.
-====== Mean of Sample Mean ======
+\begin{eqnarray*}
+E[aX] &=& a E[X] \\
+E[X+Y] &=& E[X] + E[Y] \\
+Var[aX] &=& a^{\tiny{2}} Var[X] \\
+Var[X+Y] &=& Var[X] + Var[Y]
+\end{eqnarray*}
+</WRAP>
+====== 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.
 \begin{eqnarray*}
@@ Line 14: / Line 25: @@
 \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*}
 한편, $\overline{X}$ 는
-\begin{eqnarray*}
+\begin{align*}
-\overline{X} = \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \\
+\overline{X} & = \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \\
-\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}} & = & \mu
-\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}}  & = & \frac{\sigma}{\sqrt{n}} \\
+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]  & = \frac{\sigma^2}{n} \\
+\sigma_{\overline{X}}^{2} & = \frac{\sigma^2}{n} \\
+\sigma_{\overline{X}}  & = \frac{\sigma}{\sqrt{n}} \\
+\end{align*}