corrected some pages

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

Author: JoramSoch

D/cf.md

@@ -20,6 +20,12 @@ sources:
     in: "Wikipedia, the free encyclopedia"
     pages: "retrieved on 2021-09-22"
     url: "https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)#Definition"
+  - authors: "Taboga, Marco"
+    year: 2017
+    title: "Joint characteristic function"
+    in: "Lectures on probability and mathematical statistics"
+    pages: "retrieved on 2021-10-07"
+    url: "https://www.statlect.com/fundamentals-of-probability/joint-characteristic-function"
 
 def_id: "D159"
 shortcut: "cf"

D/mgf.md

@@ -20,6 +20,12 @@ sources:
     in: "Wikipedia, the free encyclopedia"
     pages: "retrieved on 2020-01-22"
     url: "https://en.wikipedia.org/wiki/Moment-generating_function#Definition"
+  - authors: "Taboga, Marco"
+    year: 2017
+    title: "Joint moment generating function"
+    in: "Lectures on probability and mathematical statistics"
+    pages: "retrieved on 2021-10-07"
+    url: "https://www.statlect.com/fundamentals-of-probability/joint-moment-generating-function"
 
 def_id: "D2"
 shortcut: "mgf"

P/dent-addvec.md

@@ -86,7 +86,7 @@ $$ \label{eq:Y-dent-s2}
 \end{split}
 $$
 
-Using the differential $\mathrm{d}(Ax) = \left|A\right| \mathrm{d}x$, this becomes
+Using the differential $\mathrm{d}(Ax) = \lvert A \rvert \mathrm{d}x$, this becomes
 
 $$ \label{eq:Y-dent-s3}
 \begin{split}

P/dent-noninv.md

@@ -119,7 +119,7 @@ $$ \label{eq:Y-dent-s2}
 \end{split}
 $$
 
-Using the relations $y = f(x) \Rightarrow \mathrm{d}y = \left| J_f(x) \right| \, \mathrm{d}x$ and $\left|A\right|\left|B\right| = \left|AB\right|$, this becomes
+Using the relations $y = f(x) \Rightarrow \mathrm{d}y = \lvert J_f(x) \rvert \, \mathrm{d}x$ and $\lvert A \rvert \lvert B \rvert = \lvert AB \rvert$, this becomes
 
 $$ \label{eq:Y-dent-s3}
 \begin{split}
@@ -128,7 +128,7 @@ $$ \label{eq:Y-dent-s3}
 \end{split}
 $$
 
-Finally, employing [the fact](/D/pdf) that $\int_{\mathcal{X}} f_X(x) \, \mathrm{d}x = 1$ and the determinant property $\left|A^{-1}\right| = 1/\left|A\right|$, we can derive the [differential entropy](/D/dent) of $Y$ as
+Finally, employing [the fact](/D/pdf) that $\int_{\mathcal{X}} f_X(x) \, \mathrm{d}x = 1$ and the determinant property $\lvert A^{-1} \rvert = 1/\lvert A \rvert$, we can derive the [differential entropy](/D/dent) of $Y$ as
 
 $$ \label{eq:Y-dent-s4}
 \begin{split}

P/pdf-invfct.md

@@ -109,7 +109,7 @@ $$ \label{eq:A-y}
 A(y) = \left\lbrace x \in \mathbb{R}^n: g_j(x) \leq y_j \; \text{for all} \; j = 1, \ldots, n \right\rbrace
 $$
 
-and $g_j(X)$ is the function which returns the $j$-th element of $Y$, given a vector $x$.
+and $g_j(X)$ is the function which returns the $j$-th element of $Y$, given a vector $X$.
 
 <br>
 2) Next, we substitute $x = g^{-1}(y)$ into the integral which gives us