corrected some pages

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

Author: JoramSoch

D/cdf.md

@@ -34,19 +34,15 @@ 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
+1) If $X$ is a [discrete](/D/rvar-disc) [random variable](/D/rvar) with possible outcomes $\mathcal{X}$ and the [probability mass function](/D/pmf) $f_X(x)$, then [the cumulative distribution function is the function](/P/cdf-pmf) $F_X(x): \mathbb{R} \to [0,1]$ with
 
 $$ \label{eq:cdf-disc}
-F_X(x) = \sum_{\overset{z \in \mathcal{X}}{z \leq x}} f_X(z)
+F_X(x) = \sum_{\overset{t \in \mathcal{X}}{t \leq x}} f_X(t) \; .
 $$
 
-is the cumulative distribution function of $X$.
-
 <br>
-2) Let $X$ be a scalar continuous [random variable](/D/rvar) with the [probability density function](/D/pdf) $f_X(x)$. Then, the function $F_X(x): \mathbb{R} \to [0,1]$ with
+2) If $X$ is a [continuous](/D/rvar-disc) [random variable](/D/rvar) with possible outcomes $\mathcal{X}$ and the [probability density function](/D/pdf) $f_X(x)$, then [the cumulative distribution function is the function](/P/cdf-pdf) $F_X(x): \mathbb{R} \to [0,1]$ with
 
 $$ \label{eq:cdf-cont}
-F_X(x) = \int_{-\infty}^{x} f_X(z) \, \mathrm{d}z
-$$
-
-is the cumulative distribution function of $X$.
+F_X(x) = \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \; .
+$$

D/pdf.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**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
+**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 (PDF) 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](/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
+**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 (PMF) of $X$, if
 
 $$ \label{eq:pmf-def-s0}
 f_X(x) = 0

D/qf.md

@@ -27,14 +27,8 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $X$ be a [random variable](/D/rvar) with the [cumulative distribution function](/D/cdf) (CDF) $F_X(x)$. Then, the function $Q_X(p): [0,1] \to \mathbb{R}$ which is the inverse CDF
+**Definition:** Let $X$ be a [random variable](/D/rvar) with the [cumulative distribution function](/D/cdf) (CDF) $F_X(x)$. Then, the function $Q_X(p): [0,1] \to \mathbb{R}$ which is the inverse CDF is the quantile function (QF) of $X$. More precisly, the QF is the function that, for a given quantile $p \in [0,1]$, returns the smallest $x$ for which $F_X(x) = p$:
 
 $$ \label{eq:qf}
-Q_X(p) = F_X^{-1}(x)
-$$
-
-is the quantile function (QF) of $X$. More precisly, the QF is the function that, for a given quantile $p \in [0,1]$, returns the smallest $x$ for which $F_X(x) = p$:
-
-$$ \label{eq:qf-prec}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
 $$

D/rfmat.md

@@ -30,5 +30,5 @@ username: "JoramSoch"
 **Definition:** In [multiple linear regression](/D/mlr), the residual-forming matrix is the matrix $R$ that results in the vector of residuals left over by [estimated parameters](/D/emat) when right-multiplied with the measured data:
 
 $$ \label{eq:pm}
-Ry = \hat{\varepsilon} = y - \hat{y} \; .
+Ry = \hat{\varepsilon} = y - \hat{y} = y - X \hat{\beta} \; .
 $$

P/cdf-pmf.md

@@ -28,7 +28,7 @@ F_X(x) = \sum_{\overset{t \in \mathcal{X}}{t \leq x}} f_X(t) \; .
 $$
 
 
-**Proof:** The [cumulative distribution function](/D/cdf) of a [random variable](/D/rvar) $X$ is defined as the probability that $X$ is smaller than $X$:
+**Proof:** The [cumulative distribution function](/D/cdf) of a [random variable](/D/rvar) $X$ is defined as the probability that $X$ is smaller than $x$:
 
 $$ \label{eq:cdf}
 F_X(x) = \mathrm{Pr}(X \leq x) \; .

P/cuni-qf.md

@@ -30,7 +30,12 @@ $$
 Then, the [quantile function](/D/qf) of $X$ is
 
 $$ \label{eq:cuni-qf}
-Q_X(p) = bp + a(1-p) \; .
+Q_X(p) = \left\{
+\begin{array}{rl}
+-\infty \; , & \text{if} \; p = 0 \\
+bp + a(1-p) \; , & \text{when} \; p \in \left\lbrace \frac{1}{n}, \frac{2}{n}, \ldots, \frac{b-a}{n}, 1 \right\rbrace \; .
+\end{array}
+\right.
 $$
 
 
@@ -46,9 +51,15 @@ F_X(x) = \left\{
 \right.
 $$
 
-Thus, the [quantile function](/D/qf) is:
+The quantile function $Q_X(p)$ [is defined as](/D/qf) the smallest $x$, such that $F_X(x) = p$:
+
+$$ \label{eq:qf}
+Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
+$$
+
+Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, it holds that
 
-$$ \label{eq:cuni-qf-s1}
+$$ \label{eq:exp-qf-s1}
 Q_X(p) = F_X^{-1}(x) \; .
 $$
 
@@ -58,6 +69,6 @@ $$ \label{eq:cuni-cdf-s2}
 \begin{split}
 p &= \frac{x-a}{b-a} \\
 x &= p(b-a) + a \\
-x &= bp + a(1-p) = Q_X(p) \; .
+x &= bp + a(1-p) \; .
 \end{split}
 $$

P/duni-qf.md

@@ -30,7 +30,12 @@ $$
 Then, the [quantile function](/D/qf) of $X$ is
 
 $$ \label{eq:duni-qf}
-Q_X(p) = a (1-p) + (b+1) p - 1 \quad \text{where} \quad p \in \left\lbrace \frac{1}{n}, \frac{2}{n}, \ldots, \frac{b-a}{n}, 1 \right\rbrace
+Q_X(p) = \left\{
+\begin{array}{rl}
+-\infty \; , & \text{if} \; p = 0 \\
+a (1-p) + (b+1) p - 1 \; , & \text{when} \; p \in \left\lbrace \frac{1}{n}, \frac{2}{n}, \ldots, \frac{b-a}{n}, 1 \right\rbrace \; .
+\end{array}
+\right.
 $$
 
 with $n = b - a + 1$.
@@ -48,7 +53,13 @@ F_X(x) = \left\{
 \right.
 $$
 
-Because [the CDF only returns](/P/duni-cdf) multiples of $1/n$ with $n = b - a + 1$, the [quantile function](/D/qf) is only defined for such values. Since [the cumulative probability increases step-wise](/P/duni-cdf) by $1/n$ at each integer between and including $a$ and $b$, the minimum $x$ at which
+The quantile function $Q_X(p)$ [is defined as](/D/qf) the smallest $x$, such that $F_X(x) = p$:
+
+$$ \label{eq:qf}
+Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
+$$
+
+Because [the CDF only returns](/P/duni-cdf) multiples of $1/n$ with $n = b - a + 1$, the [quantile function](/D/qf) is only defined for such values. First, we have $Q_X(p) = -\infty$, if $p = 0$. Second, since [the cumulative probability increases step-wise](/P/duni-cdf) by $1/n$ at each integer between and including $a$ and $b$, the minimum $x$ at which
 
 $$ \label{eq:duni-cdf-p}
 F_X(x) = \frac{c}{n} \quad \text{where} \quad c \in \left\lbrace 1, \ldots, n \right\rbrace

P/exp-qf.md

@@ -30,7 +30,12 @@ $$
 Then, the [quantile function](/D/qf) of $X$ is
 
 $$ \label{eq:exp-qf}
-Q_X(p) = -\frac{\ln(1-p)}{\lambda} \; .
+Q_X(p) = \left\{
+\begin{array}{rl}
+-\infty \; , & \text{if} \; p = 0 \\
+-\frac{\ln(1-p)}{\lambda} \; , & \text{if} \; p > 1 \; .
+\end{array}
+\right.
 $$
 
 
@@ -45,7 +50,13 @@ F_X(x) = \left\{
 \right.
 $$
 
-Thus, the [quantile function](/D/qf) is:
+The quantile function $Q_X(p)$ [is defined as](/D/qf) the smallest $x$, such that $F_X(x) = p$:
+
+$$ \label{eq:qf}
+Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
+$$
+
+Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, it holds that
 
 $$ \label{eq:exp-qf-s1}
 Q_X(p) = F_X^{-1}(x) \; .

P/norm-qf.md

@@ -48,7 +48,7 @@ $$ \label{eq:norm-cdf}
 F_X(x) = \frac{1}{2} \left[ 1 + \mathrm{erf}\left( \frac{x-\mu}{\sqrt{2} \sigma} \right) \right] \; .
 $$
 
-Thus, the [quantile function](/D/qf) is:
+Because the CDF is strictly monotonically increasing, the [quantile function is equal to the inverse of the CDF](/P/qf-cdf):
 
 $$ \label{eq:norm-qf-s1}
 Q_X(p) = F_X^{-1}(x) \; .