From 149aef28dceb57613bfd320fd88b8b4beec1a156 Mon Sep 17 00:00:00 2001
From: ishani07 \(\sum_{i=1}^{n} (y_i - \bar{y}) = 0\).
Let’s quickly walk through the proof for this: \[\sum_{i=1}^{n} (y_i - \bar{y}) = \sum_{i=1}^{n} y_i - \sum_{i=1}^{n} \bar{y}\]
-\[ = \sum\_{i=1}^{n} y_i - n\bar{y}\]
-\[ = \sum*{i=1}^{n} y_i - n\frac{1}{n}\sum*{i=1}^{n}y\]
-\[ = \sum*{i=1}^{n} y_i - \sum*{i=1}^{n}y_i\]
+\[ = \sum_{i=1}^{n} y_i - n\bar{y}\]
+\[ = \sum_{i=1}^{n} y_i - n\frac{1}{n}\sum_{i=1}^{n}y\]
+\[ = \sum_{i=1}^{n} y_i - \sum_{i=1}^{n}y_i\]
In our calculations, we’ll also be using the definition of the variance as a sample. As a refresher:
\[\sigma_y^2 = \frac{1}{n}\sum_{i=1}^{n} (y_i - \bar{y})^2\]
Getting into our calculation for MSE minimization:
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