Merge pull request bcaffo#51 from aconley/Regression_sumdisambiguate

Improve formula display in least squares=sample mean derivation

07_RegressionModels/01_01_introduction/index.Rmd

@@ -134,19 +134,21 @@ g
 ### The math (not required for the class) follows as:
 $$ 
 \begin{align} 
-\sum_{i=1}^n (Y_i - \mu)^2 & = \
-\sum_{i=1}^n (Y_i - \bar Y + \bar Y - \mu)^2 \\ 
-& = \sum_{i=1}^n (Y_i - \bar Y)^2 + \
-2 \sum_{i=1}^n (Y_i - \bar Y)  (\bar Y - \mu) +\
-\sum_{i=1}^n (\bar Y - \mu)^2 \\
-& = \sum_{i=1}^n (Y_i - \bar Y)^2 + \
-2 (\bar Y - \mu) \sum_{i=1}^n (Y_i - \bar Y)  +\
-\sum_{i=1}^n (\bar Y - \mu)^2 \\
-& = \sum_{i=1}^n (Y_i - \bar Y)^2 + \
-2 (\bar Y - \mu)  (\sum_{i=1}^n Y_i - n \bar Y) +\
-\sum_{i=1}^n (\bar Y - \mu)^2 \\
-& = \sum_{i=1}^n (Y_i - \bar Y)^2 + \sum_{i=1}^n (\bar Y - \mu)^2\\ 
-& \geq \sum_{i=1}^n (Y_i - \bar Y)^2 \
+\sum_{i=1}^n \left(Y_i - \mu\right)^2 & = \
+\sum_{i=1}^n \left(Y_i - \bar Y + \bar Y - \mu\right)^2 \\ 
+& = \sum_{i=1}^n \left(Y_i - \bar Y\right)^2 + \
+2 \sum_{i=1}^n \left(Y_i - \bar Y\right)  \left(\bar Y - \mu\right) +\
+\sum_{i=1}^n \left(\bar Y - \mu\right)^2 \\
+& = \sum_{i=1}^n \left(Y_i - \bar Y\right)^2 + \
+2 \left(\bar Y - \mu\right) \sum_{i=1}^n \left(Y_i - \bar Y\right) +\
+\sum_{i=1}^n \left(\bar Y - \mu\right)^2 \\
+& = \sum_{i=1}^n \left(Y_i - \bar Y\right)^2 + \
+2 \left(\bar Y - \mu\right)  \left(\left(\sum_{i=1}^n Y_i\right) -\
+ n \bar Y\right) +\
+\sum_{i=1}^n \left(\bar Y - \mu\right)^2 \\
+& = \sum_{i=1}^n \left(Y_i - \bar Y\right)^2 + \
+ \sum_{i=1}^n \left(\bar Y - \mu\right)^2\\ 
+& \geq \sum_{i=1}^n \left(Y_i - \bar Y\right)^2 \
 \end{align} 
 $$