corrected some pages

Several small mistakes/errors were corrected in several proofs/definitions.

Author: JoramSoch

P/slr-mat.md

@@ -68,7 +68,7 @@ E &= (X^\mathrm{T} X)^{-1} X^\mathrm{T} \\
 &= \left( \left[ \begin{matrix} n & n\bar{x} \\ n\bar{x} & x^\mathrm{T} x \end{matrix} \right] \right)^{-1} \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\
 &= \frac{1}{n x^\mathrm{T} x - (n\bar{x})^2} \left[ \begin{matrix} x^\mathrm{T} x & -n\bar{x} \\ -n\bar{x} & n \end{matrix} \right] \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\
 &= \frac{1}{x^\mathrm{T} x - n\bar{x}^2} \left[ \begin{matrix} x^\mathrm{T} x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \left[ \begin{matrix} 1_n^\mathrm{T} \\ x^\mathrm{T} \end{matrix} \right] \\
-&= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) \, 1_n^\mathrm{T} - \bar{x} \, x^\mathrm{T} \\ - \bar{x} \, 1_n^\mathrm{T} + x^\mathrm{T} \end{matrix} \right] \; .
+&\overset{\eqref{eq:b-est-cov-den}}{=} \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) \, 1_n^\mathrm{T} - \bar{x} \, x^\mathrm{T} \\ - \bar{x} \, 1_n^\mathrm{T} + x^\mathrm{T} \end{matrix} \right] \; .
 \end{split}
 $$
 
@@ -83,7 +83,7 @@ which is an $n \times n$ matrix and can be reformulated as follows:
 
 $$ \label{eq:P-qed}
 \begin{split}
-P &= X \, E = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \left[ \begin{matrix} e_1 \\ e_2 \end{matrix} \right] \\
+P &= X \, E = \left[ 1_n, \, x \right] \left[ \begin{matrix} e_1 \\ e_2 \end{matrix} \right] \\
 &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} 1 & x_1 \\ \vdots & \vdots \\ 1 & x_n \end{matrix} \right] \left[ \begin{matrix} (x^\mathrm{T} x/n) - \bar{x} x_1 & \cdots & (x^\mathrm{T} x/n) - \bar{x} x_n \\ -\bar{x} + x_1 & \cdots & -\bar{x} + x_n \end{matrix} \right] \\
 &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) - 2 \bar{x} x_1 + x_1^2 & \cdots & (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n \\ \vdots & \ddots & \vdots \\ (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n & \cdots & (x^\mathrm{T} x/n) - 2 \bar{x} x_n + x_n^2 \end{matrix} \right] \; .
 \end{split}
@@ -101,7 +101,7 @@ which also is an $n \times n$ matrix and can be reformulated as follows:
 $$ \label{eq:R-qed}
 \begin{split}
 R &= I_n - P = \left[ \begin{matrix} 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \end{matrix} \right] - \left[ \begin{matrix} p_{11} & \cdots & p_{1n} \\ \vdots & \ddots & \vdots \\ p_{n1} & \cdots & p_{nn} \end{matrix} \right] \\
-&= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} x^\mathrm{T} x - n\bar{x}^2 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & x^\mathrm{T} x - n\bar{x}^2 \end{matrix} \right] \\
+&\overset{\eqref{eq:b-est-cov-den}}{=} \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} x^\mathrm{T} x - n\bar{x}^2 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & x^\mathrm{T} x - n\bar{x}^2 \end{matrix} \right] \\
 &- \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (x^\mathrm{T} x/n) - 2 \bar{x} x_1 + x_1^2 & \cdots & (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n \\ \vdots & \ddots & \vdots \\ (x^\mathrm{T} x/n) - \bar{x} (x_1 + x_n) + x_1 x_n & \cdots & (x^\mathrm{T} x/n) - 2 \bar{x} x_n + x_n^2 \end{matrix} \right] \\
 &= \frac{1}{(n-1)\,s_x^2} \left[ \begin{matrix} (n-1) (x^\mathrm{T} x/n) + \bar{x} (2 x_1 - n\bar{x}) - x_1^2 & \cdots & -(x^\mathrm{T} x/n) + \bar{x} (x_1 + x_n) - x_1 x_n \\ \vdots & \ddots & \vdots \\ -(x^\mathrm{T} x/n) + \bar{x} (x_1 + x_n) - x_1 x_n & \cdots &  (n-1) (x^\mathrm{T} x/n) + \bar{x} (2 x_n - n\bar{x}) - x_n^2 \end{matrix} \right] \; .
 \end{split}

P/slr-olsdist.md

@@ -39,7 +39,7 @@ $$ \label{eq:slr-olsdist}
 \left[ \begin{matrix} \hat{\beta}_0 \\ \hat{\beta}_1 \end{matrix} \right] \sim \mathcal{N}\left( \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right], \, \frac{\sigma^2}{(n-1) \, s_x^2} \cdot \left[ \begin{matrix} x^\mathrm{T}x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \right)
 $$
 
-where $s_x^2$ is the [sample variance](/D/var-samp) of $x$.
+where $\bar{x}$ is the [sample mean](/D/mean-samp) and $s_x^2$ is the [sample variance](/D/var-samp) of $x$.
 
 
 **Proof:** [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr) with

P/slr-proj.md

@@ -40,7 +40,7 @@ P\left(w \mid \hat{\beta}_0 + \hat{\beta}_1 w\right) \quad \text{with} \quad w =
 $$
 
 
-**Proof:** The intersection point of the regression line with the y-axis is
+**Proof:** The intersection point of the [regression line](/D/regline) with the y-axis is
 
 $$ \label{eq:S}
 S(0 \vert \hat{\beta}_0) \; .
@@ -89,8 +89,8 @@ With \eqref{eq:a} and \eqref{eq:b}, $w$ can be calculated as
 $$ \label{eq:w-qed}
 \begin{split}
 w &= \frac{a^\mathrm{T} b}{a^\mathrm{T} a} \\
-&= \frac{\left( \begin{matrix} 1 \\ \hat{\beta}_1 \end{matrix} \right)^\mathrm{T} \left( \begin{matrix} x_o \\ y_o - \hat{\beta}_0 \end{matrix} \right)}{\left( \begin{matrix} 1 \\ \hat{\beta}_1 \end{matrix} \right)^\mathrm{T} \left( \begin{matrix} 1 \\ \hat{\beta}_1 \end{matrix} \right)} \\
-&= \frac{x_0 + (y_o - \hat{\beta}_0) \hat{\beta}_1}{1 + \hat{\beta}_1^2}
+w &= \frac{\left( \begin{matrix} 1 \\ \hat{\beta}_1 \end{matrix} \right)^\mathrm{T} \left( \begin{matrix} x_o \\ y_o - \hat{\beta}_0 \end{matrix} \right)}{\left( \begin{matrix} 1 \\ \hat{\beta}_1 \end{matrix} \right)^\mathrm{T} \left( \begin{matrix} 1 \\ \hat{\beta}_1 \end{matrix} \right)} \\
+w &= \frac{x_0 + (y_o - \hat{\beta}_0) \hat{\beta}_1}{1 + \hat{\beta}_1^2}
 \end{split}
 $$
 
@@ -100,4 +100,4 @@ $$ \label{eq:P-qed}
 \left( \begin{matrix} x_p \\ y_p \end{matrix} \right) = \left( \begin{matrix} 0 \\ \hat{\beta}_0 \end{matrix} \right) + w \cdot \left( \begin{matrix} 1 \\ \hat{\beta}_1 \end{matrix} \right) = \left( \begin{matrix} w \\ \hat{\beta}_0 + \hat{\beta}_1 w \end{matrix} \right) \; .
 $$
 
-Together, \eqref{eq:P-qed} and \eqref{eq:w-qed} constitute the proof of \eqref{eq:slr-proj}.
+Together, \eqref{eq:P-qed} and \eqref{eq:w-qed} constitute the proof of equation \eqref{eq:slr-proj}.

P/slr-sss.md

@@ -92,12 +92,12 @@ $$ \label{eq:RSS-qed}
 \begin{split}
 \mathrm{RSS} &= \sum_{i=1}^n (y_i - \hat{y}_i)^2 \\
 &= \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2 \\
-&= \sum_{i=1}^n (y_i - \bar{y} + \hat{\beta}_1 \bar{x} - \hat{\beta}_1 x_i)^2 \\
+&\overset{\eqref{eq:slr-ols}}{=} \sum_{i=1}^n (y_i - \bar{y} + \hat{\beta}_1 \bar{x} - \hat{\beta}_1 x_i)^2 \\
 &= \sum_{i=1}^n \left( (y_i - \bar{y}) - \hat{\beta}_1 (x_i - \bar{x}) \right)^2 \\
 &= \sum_{i=1}^n \left( (y_i - \bar{y})^2 - 2 \hat{\beta}_1 (x_i - \bar{x}) (y_i - \bar{y}) + \hat{\beta}_1^2 (x_i - \bar{x})^2 \right) \\
 &= \sum_{i=1}^n (y_i - \bar{y})^2 - 2 \hat{\beta}_1 \sum_{i=1}^n (x_i - \bar{x}) (y_i - \bar{y}) + \hat{\beta}_1^2 \sum_{i=1}^n (x_i - \bar{x})^2 \\
 &= (n-1) \, s_y^2 - 2 (n-1) \, \hat{\beta}_1 \, s_{xy} + (n-1) \, \hat{\beta}_1^2 \, s_x^2 \\
-&= (n-1) \, s_y^2 - 2 (n-1) \left( \frac{s_{xy}}{s_x^2} \right) s_{xy} + (n-1) \left( \frac{s_{xy}}{s_x^2} \right)^2 s_x^2 \\
+&\overset{\eqref{eq:slr-ols}}{=} (n-1) \, s_y^2 - 2 (n-1) \left( \frac{s_{xy}}{s_x^2} \right) s_{xy} + (n-1) \left( \frac{s_{xy}}{s_x^2} \right)^2 s_x^2 \\
 &= (n-1) \, s_y^2 - (n-1) \, \frac{s_{xy}^2}{s_x^2} \\
 &= (n-1) \left( s_y^2 - \frac{s_{xy}^2}{s_x^2} \right) \; .
 \end{split}