corrected some pages

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

Author: JoramSoch

D/cdf.md

@@ -16,7 +16,7 @@ definition: "Cumulative distribution function"
 sources:
   - authors: "Wikipedia"
     year: 2020
-    title: "Probability density function"
+    title: "Cumulative distribution function"
     in: "Wikipedia, the free encyclopedia"
     pages: "retrieved on 2020-02-17"
     url: "https://en.wikipedia.org/wiki/Cumulative_distribution_function#Definition"
@@ -27,7 +27,11 @@ username: "JoramSoch"
 ---
 
 
-**Definition:**
+**Definition:** The cumulative distribution function (CDF) of a [random variable](/D/rvar) $X$ at a given value $x$ is defined as the [probability](/D/prob) that $X$ is smaller than $x$:
+
+$$ \label{eq:cdf}
+F_X(x) = \mathrm{Pr}(X \leq x) \; .
+$$
 
 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
 

D/mom.md

@@ -30,5 +30,5 @@ username: "JoramSoch"
 **Definition:** Let $X$ be a [random variable](/D/rvar) and let $n$ be a positive integer. Then, the $n$-th moment of $X$, also called ($n$-th) "raw moment" or "crude moment", is defined as the [expected value](/D/mean) of the $n$-th power of $X$:
 
 $$ \label{eq:mom}
-\mu_n = \mathrm{E}[X^n] \; .
+\mu_n' = \mathrm{E}[X^n] \; .
 $$

P/bma-lme.md

@@ -37,7 +37,7 @@ $$
 where $p(\theta \vert m_i,y)$ is the posterior distributions over $\theta$ obtained using $m_i$.
 
 
-**Proof:** According to the [law of marginal probability](/D/prob-marg), the probability of the shared parameters $\theta$ conditional on the measured data $y$ [can be obtained](/D/bma-der) by marginalizing over the discrete variable model $m$:
+**Proof:** According to the [law of marginal probability](/D/prob-marg), the probability of the shared parameters $\theta$ conditional on the measured data $y$ [can be obtained](/P/bma-der) by marginalizing over the discrete variable model $m$:
 
 $$ \label{eq:BMA-PMP}
 p(\theta|y) = \sum_{i=1}^{M} p(\theta|m_i,y) \cdot p(m_i|y) \; ,

P/ent-nonneg.md

@@ -36,23 +36,23 @@ $$
 **Proof:** The [entropy of a discrete random variable](/D/ent) is defined as
 
 $$ \label{eq:ent}
-\mathrm{H}(X) = - \sum_{i=1}^{k} p(x_i) \cdot \log_b p(x_i)
+\mathrm{H}(X) = - \sum_{x \in \mathcal{X}} p(x) \cdot \log_b p(x)
 $$
 
 The minus sign can be moved into the sum:
 
 $$ \label{eq:ent-dev}
-\mathrm{H}(X) = \sum_{i=1}^{k} \left[ p(x_i) \cdot \left( - \log_b p(x_i) \right) \right]
+\mathrm{H}(X) = \sum_{x \in \mathcal{X}} \left[ p(x) \cdot \left( - \log_b p(x) \right) \right]
 $$
 
 Because the co-domain of [probability mass functions](/D/pmf) is $[0,1]$, we can deduce:
 
 $$ \label{eq:nonneg}
 \begin{array}{rcccl}
-0 &\leq &p(x_i) &\leq &1 \\
--\infty &\leq &\log_b p(x_i) &\leq &0 \\
-0 &\leq &-\log_b p(x_i) &\leq &+\infty \\
-0 &\leq &p(x_i) \cdot \left(-\log_b p(x_i)\right) &\leq &+\infty \; .
+0 &\leq &p(x) &\leq &1 \\
+-\infty &\leq &\log_b p(x) &\leq &0 \\
+0 &\leq &-\log_b p(x) &\leq &+\infty \\
+0 &\leq &p(x) \cdot \left(-\log_b p(x)\right) &\leq &+\infty \; .
 \end{array}
 $$
 

P/gibbs-ineq.md

@@ -43,32 +43,32 @@ $$
 Let $I$ be the of all $x$ for which $p(x)$ is non-zero. Then, proving \eqref{eq:Gibbs-ineq} requires to show that
 
 $$ \label{eq:Gibbs-ineq-s1}
-\sum_{x \in I} p(x) \, \frac{\ln p(x)}{\ln q(x)} \geq 0 \; .
+\sum_{x \in I} p(x) \, \ln \frac{p(x)}{q(x)} \geq 0 \; .
 $$
 
 Because $\ln x \leq x - 1$, i.e. $-\ln x \geq 1 - x$, for all $x > 0$, with equality only if $x = 1$, we can say about the left-hand side that
 
 $$ \label{eq:Gibbs-ineq-s2}
 \begin{split}
-\sum_{x \in I} p(x) \, \frac{\ln p(x)}{\ln q(x)} &\geq \sum_{x \in I} p(x) \left( 1 - \frac{p(x)}{q(x)} \right) \\
+\sum_{x \in I} p(x) \, \ln \frac{p(x)}{q(x)} &\geq \sum_{x \in I} p(x) \left( 1 - \frac{p(x)}{q(x)} \right) \\
 &= \sum_{x \in I} p(x) - \sum_{x \in I} q(x) \; .
 \end{split}
 $$
 
-Finally, since $p(x)$ and $q(x)$ are [probability mass functions](/D/pmf), it holds that $0 \leq p(x),q(x) \leq 1$ and we also have
+Finally, since $p(x)$ and $q(x)$ are [probability mass functions](/D/pmf), we have
 
 $$ \label{eq:p-q-pmf}
 \begin{split}
-\sum_{x \in I} p(x) &= 1 \\
-\sum_{x \in I} q(x) &\leq 1 \; ,
+0 \leq p(x) \leq 1, \quad \sum_{x \in I} p(x) &= 1 \quad \text{and} \\
+0 \leq q(x) \leq 1, \quad \sum_{x \in I} q(x) &\leq 1 \; ,
 \end{split}
 $$
 
 such that it follows from \eqref{eq:Gibbs-ineq-s2} that
 
 $$ \label{eq:Gibbs-ineq-s3}
 \begin{split}
-\sum_{x \in I} p(x) \, \frac{\ln p(x)}{\ln q(x)} &\geq \sum_{x \in I} p(x) - \sum_{x \in I} q(x) \\
+\sum_{x \in I} p(x) \, \ln \frac{p(x)}{q(x)} &\geq \sum_{x \in I} p(x) - \sum_{x \in I} q(x) \\
 &= 1 - \sum_{x \in I} q(x) \geq 0 \; .
 \end{split}
 $$

P/kl-nonneg.md

@@ -39,13 +39,13 @@ with $\mathrm{KL}[P \vert \vert Q] = 0$, if and only if $P = Q$.
 **Proof:** The discrete [Kullback-Leibler divergence](/D/kl) is defined as
 
 $$ \label{eq:KL}
-\mathrm{KL}[P||Q] = \sum_{x \in \mathcal{X}} p(x) \cdot \log \frac{p(x)}{q(x)} \, \mathrm{d}x
+\mathrm{KL}[P||Q] = \sum_{x \in \mathcal{X}} p(x) \cdot \log \frac{p(x)}{q(x)}
 $$
 
 which can be reformulated into
 
 $$ \label{eq:KL-dev}
-\mathrm{KL}[P||Q] = \sum_{x \in \mathcal{X}} p(x) \cdot \log p(x) - \sum_{x \in \mathcal{X}} p(x) \cdot \log q(x) \, \mathrm{d}x \; .
+\mathrm{KL}[P||Q] = \sum_{x \in \mathcal{X}} p(x) \cdot \log p(x) - \sum_{x \in \mathcal{X}} p(x) \cdot \log q(x) \; .
 $$
 
 [Gibbs' inequality](/P/gibbs-ineq) states that the [entropy](/D/ent) of a probability distribution is always less than or equal to the [cross-entropy](/D/ent-cross) with another probability distribution – with equality only if the distributions are identical –,