corrected some pages

Several small corrections were done to several proofs and definitions.

Author: JoramSoch

P/ind-self.md

@@ -49,7 +49,7 @@ P(E)   &= \left( P(E) \right)^2 \; .
 \end{split}
 $$
 
-This is only fulfilled, if
+For $0 \leq P(E) \leq 1$, this is only fulfilled, if
 
 $$ \label{eq:ind-self-qed}
 P(E) = 0 \quad \text{or} \quad P(E) = 1 \; .

P/mean-mse.md

@@ -43,7 +43,7 @@ $$
 Setting the first derivative
 
 $$ \label{eq:dmse-da}
-\frac{d}{da} \left[ a^2 - 2a\mu + \mathrm{E}(X_i^2) \right] = 2a - 2\mu
+\frac{\mathrm{d}}{\mathrm{d}a} \left[ a^2 - 2a\mu + \mathrm{E}(X_i^2) \right] = 2a - 2\mu
 $$
 
 to zero to perform a derivative test, we obtain:

P/mlr-fomnibus.md

@@ -62,7 +62,7 @@ under the [null hypothesis](/D/h0) that all [regression coefficients](/D/mlr) $\
 
 $$ \label{eq:mlr-f-omnibus-h0}
 H_0: \; \beta_1 = 0_{p_1} \quad \Leftrightarrow \quad
-\beta_j = 0 \quad \text{for all} \quad j = (p_0+1),\ldots,p \; .
+\beta_{1j} = 0 \quad \text{for all} \quad j=1,\ldots,p_1 \; .
 $$
 
 In \eqref{eq:mlr-f-omnibus}, $\hat{\varepsilon}$ and $\hat{\varepsilon}_0$ are the [residual vectors](/P/mlr-mat) when using either the full design matrix $X$ or the reduced design matrix $X_0$:

P/mlr-llr.md

@@ -15,7 +15,7 @@ theorem: "Log-likelihood ratio"
 
 sources:
   - authors: "Ostwald, Dirk"
-    year: 2024
+    year: 2023
     title: "F-Statistiken"
     in: "Allgemeines Lineares Modell"
     pages: "Einheit (8), Folien 20-22"

P/mlr-olsdist.md

@@ -27,6 +27,18 @@ sources:
     in: "Mathematics for Brain Imaging"
     pages: "ch. 1.5, pp. 39-41, eqs. 1.106-1.110"
     url: "https://ueapsylabs.co.uk/sites/wpenny/mbi/mbi_course.pdf"
+  - authors: "Ostwald, Dirk"
+    year: 2023
+    title: "Modellformulierung"
+    in: "Allgemeines Lineares Modell"
+    pages: "Einheit (5), Folie 14"
+    url: "https://www.ipsy.ovgu.de/ipsy_media/Methodenlehre+I/Sommersemester+2023/Allgemeines+Lineares+Modell/5_Modellformulierung.pdf"
+  - authors: "Ostwald, Dirk"
+    year: 2023
+    title: "Parameterschätzung"
+    in: "Allgemeines Lineares Modell"
+    pages: "Einheit (6), Folien 10-12"
+    url: "https://www.ipsy.ovgu.de/ipsy_media/Methodenlehre+I/Sommersemester+2023/Allgemeines+Lineares+Modell/6_Parametersch%C3%A4tzung.pdf"
 
 proof_id: "P400"
 shortcut: "mlr-olsdist"