corrected some pages

Several small corrections were done to several proofs and definitions.

Author: JoramSoch

D/ent.md

@@ -34,4 +34,4 @@ $$ \label{eq:ent}
 \mathrm{H}(X) = - \sum_{x \in \mathcal{X}} p(x) \cdot \log_b p(x)
 $$
 
-where $b$ is the base of the logarithm specifying in which unit the entropy is determined.
+where $b$ is the base of the logarithm specifying in which unit the entropy is determined. By convention, $0 \cdot \log 0$ is taken to be zero when calculating the entropy of $X$.

D/kl.md

@@ -43,4 +43,6 @@ $$ \label{eq:KL-cont}
 \mathrm{KL}[P||Q] = \int_{\mathcal{X}} p(x) \cdot \log \frac{p(x)}{q(x)} \, \mathrm{d}x
 $$
 
-where $p(x)$ and $q(x)$ are the [probability density functions](/D/pdf) of $P$ and $Q$.
+where $p(x)$ and $q(x)$ are the [probability density functions](/D/pdf) of $P$ and $Q$.
+
+[By convention](/D/ent), $0 \cdot \log 0$ is taken to be zero when calculating the divergence between $P$ and $Q$.

P/bvn-pdf.md

@@ -40,7 +40,7 @@ $$ \label{eq:mvn-pdf}
 f_X(x) = \frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \cdot \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right] \; .
 $$
 
-Plugging in $n = 2$ as well as $\mu$ and $\Sigma$ from equation \eqref{eq:bvn}, we obtain:
+Plugging in $n = 2$, $\mu = \left[ \begin{matrix} \mu_1 \\\\ \mu_2 \end{matrix} \right]$ and $\Sigma = \left[ \begin{matrix} \sigma_1^2 \& \sigma_{12} \\\\ \sigma_{12} \& \sigma_2^2 \end{matrix} \right]$, we obtain:
 
 $$ \label{eq:bvn-pdf-s1}
 \begin{split}

P/bvn-pdfcorr.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $X = \left[ \begin{matrix} X_1 & X_2 \end{matrix} \right]^\mathrm{T}$ follow a bivariate normal distribution:
+**Theorem:** Let $X = \left[ \begin{matrix} X_1 \\\\ X_2 \end{matrix} \right]$ follow a bivariate normal distribution:
 
 $$ \label{eq:bvn}
 X \sim \mathcal{N}\left( \mu = \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], \Sigma = \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \right) \; .

P/cuni-kl.md

@@ -57,7 +57,7 @@ $$ \label{eq:cuni-KL-s2}
 \mathrm{KL}[P\,||\,Q] = \int_{-\infty}^{a_1} 0 \cdot \ln \frac{0}{q(x)} \, \mathrm{d}x + \int_{a_1}^{b_1} p(x) \, \ln \frac{p(x)}{q(x)} \, \mathrm{d}x + \int_{b_1}^{+\infty} 0 \cdot \ln \frac{0}{q(x)} \, \mathrm{d}x \; .
 $$
 
-Now, $(0 \cdot \ln 0)$ is taken to be $0$ [by convention](/D/ent), such that
+Now, $(0 \cdot \ln 0)$ is taken to be [zero by convention](/D/ent), such that
 
 $$ \label{eq:cuni-KL-s3}
 \mathrm{KL}[P\,||\,Q] = \int_{a_1}^{b_1} p(x) \, \ln \frac{p(x)}{q(x)} \, \mathrm{d}x

P/f-pdf.md

@@ -30,7 +30,7 @@ username: "JoramSoch"
 **Theorem:** Let $F$ be a [random variable](/D/rvar) following an [F-distribution](/D/f):
 
 $$ \label{eq:f}
-F \sim F(u,v) \; .
+F \sim \mathrm{F}(u,v) \; .
 $$
 
 Then, the [probability density function](/D/pdf) of $F$ is
@@ -43,7 +43,7 @@ $$
 **Proof:** An [F-distributed random variable](/D/f) is defined as the ratio of two [chi-squared random variables](/D/chi2), divided by their [degrees of freedom](/D/dof)
 
 $$ \label{eq:f-def}
-X \sim \chi^2(u), \; Y \sim \chi^2(v) \quad \Rightarrow \quad F = \frac{X/u}{Y/v} \sim F(u,v)
+X \sim \chi^2(u), \; Y \sim \chi^2(v) \quad \Rightarrow \quad F = \frac{X/u}{Y/v} \sim \mathrm{F}(u,v)
 $$
 
 where $X$ and $Y$ are [independent of each other](/D/ind).