Differences

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--- sand_box [2024/05/23 08:57] – hkimscil
+++ sand_box [2025/09/10 11:46] (current) – hkimscil
@@ Line 1: / Line 1: @@
+\begin{eqnarray*}
+& & \frac{3}{4 \pi} \sqrt{4 \cdot x^2 12} \\
+& & \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} \\
+& & {\it f}(x) = \frac{1}{\sqrt{x} x^2} \\
+& & e^{i \pi} + 1 = 0\;
+\end{eqnarray*}
+<latex>\setlength{\unitlength}{1mm} \begin{picture}(93,46) \put( 0,14){\vector(1,0){60}} \put(61,13){$x$} \put(20,4){\vector(0,1){37}} \put(19,43){$y$} \put(50,34){\circle*{2}} \put(52,35){$P$} \multiput(20,34)(4,0){8}{\line(1,0){2}} \put(14.5,33.5){$y_P$} \multiput(50,14)(0,4){5}{\line(0,1){2}} \put(48,11){$x_P$} \put( 2,8){\vector(3,1){56}} \put(59,26.5){$x'$} \multiput(50,34)(1.9,-5.7){2} {\line(1,-3){1.2}} \put(52,22){$x_P'$} \multiput(50,34)(-5.8,-1.933){6} {\line(-3,-1){3.6}} \put(12,21){$y_P'$} \put(22,8){\vector(-1,3){10.5}} \put(10,41){$y'$} \end{picture}</latex>
 <WRAP tabs>
   * [[:sand box/intro]]
@@ Line 5: / Line 16: @@
 </WRAP>
+{{tabinclude>page1|Top page,page2|Second page,*page3}}
+<tabbed>
+  * page1|top page
+  * page2
+  * *page3
+</tabbed>
 [{{:r.regressionline3.png}}]
-\begin{eqnarray*}
+\begin{align*}
-&\sum{(Y_i - \hat{Y_i})^2} \\
+& \;\;\;\; \sum{(Y_i - \hat{Y_i})^2} \\
 &= \sum{(Y_i - (a + bX_i))^2}  \;\;\; \because \hat{Y_i} = a + bX_i \\
 &= \text{SSE or SS.residual} \;\;\; \text{(and this should be the least value.)} \\
-\end{eqnarray*}
+\end{align*}
 <WRAP box>
-\begin{eqnarray*}
+\begin{align*}
-\text{for a (constant)} \\
+&\text{for a (constant)} \\ \\
-\\
+&\dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2} \\
-\dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2} & = & \sum \dfrac{\text{d}}{\text{dv}} {(Y_i - (a + bX_i))^2} \\
+&=  \sum \dfrac{\text{d}}{\text{dv}} {(Y_i - (a + bX_i))^2} \\
-& = & \sum{2 (Y_i - (a + bX_i))} * (-1) \;\;\;\; \\
+&= \sum{2 (Y_i - (a + bX_i))} * (-1) \;\;\;\; \\
-& \because & \dfrac{\text{d}}{\text{dv for a}} (Y_i - (a+bX_i)) = -1 \\
+&\because \dfrac{\text{d}}{\text{dv for a}} (Y_i - (a+bX_i)) = -1 \\
-& = & -2 \sum{(Y_i - (a + bX_i))} \\
+& = -2 \sum{(Y_i - (a + bX_i))} \\
 \\
-\text{in order to have the least value, the above should be zero} \\
+&\text{in order to have the least value, the above should be zero} \\
 \\
--2 \sum{(Y_i - (a + bX_i))} & = &  0 \\
+&-2 \sum{(Y_i - (a + bX_i))} = 0 \\
-\sum{(Y_i - (a + bX_i))} & = & 0 \\
+&\sum{(Y_i - (a + bX_i))} = 0 \\
-\sum{Y_i} - \sum{a} - b \sum{X_i} & = & 0 \\
+&\sum{Y_i} - \sum{a} - b \sum{X_i} = 0 \\
-\sum{Y_i} - n*{a} - b \sum{X_i} & = & 0 \\
+&\sum{Y_i} - n*{a} - b \sum{X_i} = 0 \\
-n*{a} & = & \sum{Y_i} - b \sum{X_i} \\
+&n*{a} = \sum{Y_i} - b \sum{X_i} \\
-a & = & \dfrac{\sum{Y_i}}{n} - b \dfrac{\sum{X_i}}{n} \\
+&a = \dfrac{\sum{Y_i}}{n} - b \dfrac{\sum{X_i}}{n} \\
-a & = & \overline{Y} - b \overline{X} \\
+&a = \overline{Y} - b \overline{X} \\
-\end{eqnarray*}
+\end{align*}
 </WRAP>