Differences

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--- estimated_standard_deviation [2017/12/11 09:41] – hkimscil
+++ estimated_standard_deviation [2020/04/09 01:37] – hkimscil
@@ Line 1: / Line 1: @@
 ====== Why n-1 ======
+우선,
+\begin{eqnarray*}
+Var[X] & = & {E{(X-\mu)^2}} \\
+& = & E[(X^2 - 2 X \mu + \mu^2)] \\
+& = & E[X^2] - 2 \mu E[X] + E[\mu^2] \\
+& = & E[X^2] - 2 \mu E[X] + E[\mu^2], \;\; \text{because E[X]=} \mu \text{, \; E[} \mu^2 \text{] = } \mu^2, \\
+& = & E[X^2] - 2 \mu^2 + \mu^2   \\
+& = & E[X^2] - \mu^2
+\end{eqnarray*}
+이므로
+\begin{eqnarray*}
+E[X^2] & = & \sigma^{2} - [\mu]^2 \;\;\; \dots\dots\dots\dots\dots [1]
+\end{eqnarray*}
+마찬가지로
+\begin{eqnarray*}
+Var[\overline{X}] & = & E[\overline{X}^2] - [E(\overline{X})]^2 \\
+& = & E[\overline{X}^2] - [\mu]^2
+\end{eqnarray*}
+따라서
+\begin{eqnarray*}
+E[\overline{X}^{2}] & = & \frac {\sigma^{2}} {n} - [\mu]^2 \;\;\; \dots\dots\dots\dots\dots [2]
+\end{eqnarray*}
 Why we use n-1 instead of n in getting standard deviation \\
 http://www.qc.edu.hk/math/Advanced%20Level/Standard_deviation.htm \\
@@ Line 118: / Line 150: @@
 $$
 \begin{align*}
- Var [X + Y] & =  E[(X^2 + 2XY + Y^2)] - (a^2 - 2ab - b^2) \\
+ Var [X + Y] & =  E[(X^2 + 2XY + Y^2)] - (a^2 + 2ab - b^2) \\
  & = E[X^2] - a^2 + E[Y^2] - b^2 \\
  & = Var[X] + Var[Y]