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--- correlation [2023/10/05 16:20] – [e.g. 1,] hkimscil
+++ correlation [2023/10/05 17:19] (current) – [e.g. 1,] hkimscil
@@ Line 154: / Line 154: @@
 \end{eqnarray}
-----
+<WRAP box>
 그런데 왜 다음과 같은 공식인지는
 \begin{align}
@@ Line 167: / Line 167: @@
 & \text{is n instead of n-1} \nonumber \\
 & \text{And we also know that} \nonumber \\
-Var[X] & = E[X^2] − (E[X])^2 \;\; \text{see} \ref{expected_value_and_variance_properties#mjx-eqn-var.theorem.1} \nonumber \\
+Var[X] & = E[X^2] − (E[X])^2 \;\; \nonumber \\
 & = \frac {\Sigma {X^2}}{n} - \left(\frac{\Sigma{X}}{n} \right)^2 \nonumber \\
 & = \frac {\Sigma {X^2}}{n} - \frac{(\Sigma{X})^2}{n^2} \nonumber \\
@@ Line 173: / Line 173: @@
 SS_{\small{X}} & = \Sigma {X^2} - \frac{(\Sigma{X})^2}{n}  \;\;\;\;\; \text{That is,  } \; \ref{ss.simplified} \nonumber \\
 \end{align}
+</WRAP>
+<WRAP box>
+또한
+\begin{align}
+SP & = & \sum XY - \frac{\sum X \sum Y}{n} \label{sp.simplified} \tag{SP simplified} \\
+\end{align}
+\begin{align}
+Cov[X,Y] & = E[(X-\overline{X})(Y-\overline{Y})] \nonumber \\
+ & = E[XY - X \overline{Y} - \overline{X} Y - \overline{X} \overline{Y}] \nonumber \\
+ & = E[XY] - E[X] \overline{Y} - \overline{X} E[Y] + \overline{X} \overline{Y} \nonumber \\
+ & \because \;\;\; E[c] = c \;\;\; \text{and, }  \overline{X} = E[X] \nonumber \\
+ & =  E[XY] - E[X]E[Y] - E[X]E[Y] + E[X]E[Y] \nonumber \\
+ & =  E[XY] - E[X]E[Y] \nonumber \\
+ & =  \frac{\Sigma{XY}}{n} - \frac{\Sigma{X}}{n} \frac{\Sigma{Y}}{n}  \nonumber \\
+ & \therefore  \nonumber \\
+SP & = \Sigma{XY} - \frac{\Sigma{X} \Sigma{Y}}{n}  \;\;\;\;\; \text{That is,  } \; \ref{sp.simplified} \nonumber \\
+\end{align}
+</WRAP>
 이제 r (correlation coefficient) 값은: