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--- variance [2025/09/30 14:09] – [다른 공식] hkimscil
+++ variance [2025/09/30 14:23] (current) – [Read more] hkimscil
@@ Line 64: / Line 64: @@
 <BOOKMARK:variance_cal>
-$ \sigma^2 = \displaystyle \frac{\displaystyle \sum (X_i-\mu)^2}{N}$ 에서
+$ \sigma^2 = \displaystyle \frac {\displaystyle \sum (X_i-\mu)^2}{N}$ 에서
-\begin{eqnarray*}
+\begin{eqnarray}
-\sum (X_i-\mu)^2 & = & \sum [(X_i^2)-(2*X_i*\mu)+(\mu^2)] \\
+\sum (X_i-\mu)^2 & = & \sum [(X_i^2)-(2*X_i*\mu)+(\mu^2)] \nonumber \\
-& = & \sum (X_i^2) - \sum (2*X_i*\mu) + \sum (\mu^2) \\
+& = & \sum (X_i^2) - \sum (2*X_i*\mu) + \sum (\mu^2) \nonumber \\
-& = & \sum (X_i^2) - 2 \mu \sum (X_i) + N (\mu^2) \\
+& = & \sum (X_i^2) - 2 \mu \sum (X_i) + N (\mu^2) \nonumber \\
-& = & \sum (X_i^2) - 2 \mu (N * \mu) + N (\mu^2) \\
+& = & \sum (X_i^2) - 2 \mu (N * \mu) + N (\mu^2) \nonumber \\
 & = & \sum (X_i^2) - N * \mu^2
-\end{eqnarray*}
+\end{eqnarray}
 위에서, $\text{2 and}$ $\mu$ $\text{are constants. }$
@@ Line 79: / Line 79: @@
 \begin{eqnarray}
-\sigma^2 & = & \displaystyle \frac{\sum (X_i^2) - N * \mu^2}{N} \nonumber \\
+\sigma^2 & = & \displaystyle{\frac {\text{(1)}}{N}} \nonumber \\
-& = & \displaystyle \frac{\sum (X_i^2)}{N} - \mu^2
+& = & \displaystyle \frac{\sum (X_i^2) - N * \mu^2}{N} \nonumber \\
-& = & E(X-\mu)^{2} = E(X^{2})-E(X)^{2} \;\;\;
+& = & \displaystyle \frac{\sum (X_i^2)}{N} - \mu^2 \\
+& = & E(X^{2})-E(X)^{2} \;\;\;
 \end{eqnarray}
@@ Line 102: / Line 103: @@
 <code>
+> set.seed(1)
 > a <- rnorm2(100000000, 100, 10)
 > a.mean <- mean(a)
@@ Line 113: / Line 115: @@
 </code>
-See also [[Standard Deviation]] \\
+더 자세한 것은 [[:why n-1]] 참조. \\