corrected some pages

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

Author: JoramSoch

P/mvn-ind.md

@@ -69,12 +69,31 @@ $$ \label{eq:x-ind-dev}
 \end{split}
 $$
 
-which, given the laws for matrix determinants and matrix inverses, is only fulfilled if
+which is only fulfilled by a diagonal covariance matrix
 
 $$ \label{eq:Sigma-diag-qed}
-\Sigma = \mathrm{diag}\left( \left[ \sigma^2_1, \ldots, \sigma^2_n \right] \right) \; .
+\Sigma = \mathrm{diag}\left( \left[ \sigma^2_1, \ldots, \sigma^2_n \right] \right) \; ,
+$$
+
+because the determinant of a diagonal matrix is a product
+
+$$ \label{eq:diag-det}
+| \mathrm{diag}\left( \left[ a_1, \ldots, a_n \right] \right) | = \prod_{i=1}^n a_i \; ,
 $$
 
+the inverse of a diagonal matrix is a diagonal matrix
+
+$$ \label{eq:diag-inv}
+\mathrm{diag}\left( \left[ a_1, \ldots, a_n \right] \right)^{-1} = \mathrm{diag}\left( \left[ 1/a_1, \ldots, 1/a_n \right] \right)
+$$
+
+and the squared form with a diagonal matrix is
+
+$$ \label{eq:diag-sqr}
+x^\mathrm{T} \mathrm{diag}\left( \left[ a_1, \ldots, a_n \right] \right) x = \sum_{i=1}^n a_i x_i^2 \; .
+$$
+
+
 <br>
 2) Let
 

P/ug-ttest2.md

@@ -48,10 +48,10 @@ $$ \label{eq:t}
 t = \frac{(\bar{y}_1-\bar{y}_2)-\mu_\Delta}{s_p \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}
 $$
 
-with [sample means](/D/mean-samp) $\bar{y}_1$ and $\bar{y}_2$ and [pooled standard deviation](/D/std-pool) $s_p$ follows a [Student's t-distribution](/D/t) with $n_1+n_2-1$ [degrees of freedom](/D/dof)
+with [sample means](/D/mean-samp) $\bar{y}_1$ and $\bar{y}_2$ and [pooled standard deviation](/D/std-pool) $s_p$ follows a [Student's t-distribution](/D/t) with $n_1+n_2-2$ [degrees of freedom](/D/dof)
 
 $$ \label{eq:t-dist}
-t \sim \mathrm{t}(n_1+n_2-1)
+t \sim \mathrm{t}(n_1+n_2-2)
 $$
 
 under the [null hypothesis](/D/h0)