corrected some pages

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

Author: JoramSoch

D/ind.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Generally speaking, [random variables](/D/rvar) are statistically independent, if their [joint probability](/D/prob-joint) can be expressed in terms of their [marginal probability](/D/prob-marg).
+**Definition:** Generally speaking, [random variables](/D/rvar) are statistically independent, if their [joint probability](/D/prob-joint) can be expressed in terms of their [marginal probabilities](/D/prob-marg).
 
 <br>
 1) A set of discrete [random variables](/D/rvar) $X_1, \ldots, X_n$ with possible values $\mathcal{X}_1, \ldots, \mathcal{X}_n$ is called statistically independent, if
@@ -45,7 +45,7 @@ $$ \label{eq:cont-ind-F}
 F_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = \prod_{i=1}^{n} F_{X_i}(x_i) \quad \text{for all} \; x_i \in \mathcal{X}_i, \; i = 1, \ldots, n
 $$
 
-or equivalently, if the [probability densities](/D/pdf) exist,
+or equivalently, if the [probability densities](/D/pdf) exist, if
 
 $$ \label{eq:cont-ind-f}
 f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = \prod_{i=1}^{n} f_{X_i}(x_i) \quad \text{for all} \; x_i \in \mathcal{X}_i, \; i = 1, \ldots, n

D/pdf.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $X$ be a continuous [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to \mathbb{R}$ is the probability density function of $X$, if
+**Definition:** Let $X$ be a [continuous](/D/rvar-disc) [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to \mathbb{R}$ is the probability density function of $X$, if
 
 $$ \label{eq:pdf-def-s0}
 f_X(x) \geq 0

D/pmf.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $X$ be a discrete [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to [0,1]$ is the probability mass function of $X$, if
+**Definition:** Let $X$ be a [discrete](/D/rvar-disc) [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to [0,1]$ is the probability mass function of $X$, if
 
 $$ \label{eq:pmf-def-s0}
 f_X(x) = 0

P/gam-cdf.md

@@ -14,6 +14,12 @@ topic: "Gamma distribution"
 theorem: "Cumulative distribution function"
 
 sources:
+  - authors: "Wikipedia"
+    year: 2020
+    title: "Incomplete gamma function"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2020-10-29"
+    url: "https://en.wikipedia.org/wiki/Incomplete_gamma_function#Definition"
 
 proof_id: "P178"
 shortcut: "gam-cdf"

P/gam-sgam.md

@@ -66,21 +66,21 @@ $$
 
 Applying \eqref{eq:cdf-sifct} to \eqref{eq:gam-cdf}, we have:
 
-\begin{equation} \label{eq:Y-cdf-s1}
+$$ \label{eq:Y-cdf-s1}
 \begin{split}
 F_Y(y) &\overset{\eqref{eq:cdf-sifct}}{=} F_X(g^{-1}(y)) \\
 &\overset{\eqref{eq:gam-cdf}}{=} \int_{-\infty}^{y/b} \frac{b^a}{\Gamma(a)} t^{a-1} \exp[-b t] \, \mathrm{d}t \; .
 \end{split}
-\end{equation}
+$$
 
 Substituting $s = b t$, such that $t = s/b$, we obtain
 
-\begin{equation} \label{eq:Z-cdf-s2}
+$$ \label{eq:Z-cdf-s2}
 \begin{split}
 F_Y(y) &= \int_{-b \infty}^{b (y/b)} \frac{b^a}{\Gamma(a)} \left(\frac{s}{b}\right)^{a-1} \exp\left[-b \left(\frac{s}{b}\right)\right] \, \mathrm{d}\left(\frac{s}{b}\right) \\
 &= \int_{-\infty}^{y} \frac{b^a}{\Gamma(a)} \, \frac{1}{b^{a-1} \, b} \, s^{a-1} \exp[-s] \, \mathrm{d}s \\
 &= \int_{-\infty}^{y} \frac{1}{\Gamma(a)} s^{a-1} \exp[-s] \, \mathrm{d}s
 \end{split}
-\end{equation}
+$$
 
 which is the [cumulative distribution function](/D/cdf) of the [standard gamma distribution](/D/sgam).

P/pdf-sifct.md

@@ -73,12 +73,12 @@ $$
 
 2) If $y$ belongs to the support of $Y$, then $f_Y(y)$ can be derived using the chain rule:
 
-\begin{equation} \label{eq:pdf-sifct-p2}
+$$ \label{eq:pdf-sifct-p2}
 \begin{split}
 f_Y(y) &\overset{\eqref{eq:pdf-cdf}}{=} \frac{\mathrm{d}}{\mathrm{d}y} F_Y(y) \\
 &\overset{\eqref{eq:cdf-sifct}}{=} \frac{\mathrm{d}}{\mathrm{d}y} F_X(g^{-1}(y)) \\
 &= f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; .
 \end{split}
-\end{equation}
+$$
 
 Taking together \eqref{eq:pdf-sifct-p1} and \eqref{eq:pdf-sifct-p2}, eventually proves \eqref{eq:pdf-sifct}.