mean_and_variance_of_the_sample_mean
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mean_and_variance_of_the_sample_mean [2020/12/05 18:31] – [Mean of the 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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한편, $\overline{X}$ 는 | 한편, $\overline{X}$ 는 | ||
- | |||
- | |||
\begin{align*} | \begin{align*} | ||
\overline{X} & = \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \\ | \overline{X} & = \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \\ | ||
- | \\ | + | \end{align*} |
- | \\ | + | 이고 |
+ | |||
+ | \begin{align*} | ||
E\left[\overline{X}\right] & = 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)\\ | ||
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====== Variance of the sample mean ====== | ====== Variance of the sample mean ====== | ||
- | \begin{eqnarray*} | + | \begin{align*} |
- | Var[\overline{X}] & = & Var \left[ \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \right] \\ | + | Var\left[\overline{X}\right] & = Var \left[ \dfrac {X_{1} + X_{2} + . . . + X_{n}} {n} \right] \\ |
- | & = & (\frac{1}{n})^2 Var \left[ X_{1} + X_{2} + . . . + X_{n} \right] \\ | + | & = \left(\frac{1}{n}\right)^2 Var \left[ X_{1} + X_{2} + . . . + X_{n} \right] \\ |
- | & = & (\frac{1}{n})^2 (Var[X_{1} + X_{2} + . . . + X_{n}]) \\ | + | & = \left(\frac{1}{n}\right)^2 \left(Var[X_{1} + X_{2} + . . . + X_{n}]\right) \\ |
- | & = & (\frac{1}{n})^2 (Var[X_{1}] + Var[X_{2}] + . . . + Var[X_{n}])\\ | + | & = \left(\frac{1}{n}\right)^2 \left(Var[X_{1}] + Var[X_{2}] + . . . + Var[X_{n}]\right)\\ |
- | & = & (\frac{1}{n})^2 (\sigma^2 + \sigma^2 + . . . + \sigma^2) \\ | + | & = \left(\frac{1}{n}\right)^2 \left(\sigma^2 + \sigma^2 + . . . + \sigma^2\right) \\ |
- | & = & \frac{1}{n^2} n \sigma^2 \\ | + | & = \frac{1}{n^2} n \sigma^2 \\ |
- | & = & \frac{\sigma^2}{n} | + | & = \frac{\sigma^2}{n} |
- | \end{eqnarray*} | + | \\ |
+ | \\ | ||
+ | Var\left[\overline{X}\right] | ||
+ | \sigma_{\overline{X}}^{2} & = \frac{\sigma^2}{n} \\ | ||
+ | \sigma_{\overline{X}} | ||
+ | \end{align*} | ||
- | \begin{eqnarray*} | ||
- | Var[\overline{X}] & = & \frac{\sigma^2}{n} \\ | ||
- | \sigma_{\overline{X}}^{2} & = & \frac{\sigma^2}{n} \\ | ||
- | \sigma_{\overline{X}} | ||
- | \end{eqnarray*} |
mean_and_variance_of_the_sample_mean.1607160695.txt.gz · Last modified: 2020/12/05 18:31 by hkimscil