corrected some pages

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

Author: JoramSoch

D/cdf.md

@@ -33,6 +33,7 @@ $$ \label{eq:cdf}
 F_X(x) = \mathrm{Pr}(X \leq x) \; .
 $$
 
+<br>
 1) Let $X$ be a discrete [random variable](/D/rvar) with possible outcomes $\mathcal{X}$ and the [probability mass function](/D/pmf) $f_X(x)$. Then, the function $F_X(x): \mathbb{R} \to [0,1]$ with
 
 $$ \label{eq:cdf-disc}

P/kl-nonneg2.md

@@ -50,12 +50,12 @@ $$
 
 where $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ be non-negative real numbers and $a = \sum_{i=1}^{n} a_i$ and $b = \sum_{i=1}^{n} b_i$. Because $p(x)$ and $q(x)$ are [probability mass functions](/D/pmf), such that
 
-\begin{equation} \label{eq:p-q-pmf}
+$$ \label{eq:p-q-pmf}
 \begin{split}
 p(x) \geq 0, \quad \sum_{x \in \mathcal{X}} p(x) &= 1 \quad \text{and} \\
 q(x) \geq 0, \quad \sum_{x \in \mathcal{X}} q(x) &= 1 \; ,
 \end{split}
-\end{equation}
+$$
 
 theorem \eqref{eq:KL-nonneg} is simply a special case of \eqref{eq:logsum-ineq}, i.e.
 

P/momcent-1st.md

@@ -42,11 +42,11 @@ $$
 
 Due to the [linearity of the expected value](/P/mean-lin) and by plugging in $\mu = \mathrm{E}(X)$, we have
 
-\begin{equation} \label{eq:momcent-1st-qed}
+$$ \label{eq:momcent-1st-qed}
 \begin{split}
 \mu_1 &= \mathrm{E}\left[ X-\mu \right] \\
 &= \mathrm{E}(X) - \mu \\
 &= \mathrm{E}(X) - \mathrm{E}(X) \\
 &= 0 \; .
 \end{split}
-\end{equation}
+$$