The number of possible combinations of $k$ elements from $n$ elements is
given by
\[
{n\choose k}=\frac{n!}{k!(n-k)!}
\]
The number of permutations of $p$ from $n$ is given by
\[
\frac{n!}{(n-p)!}=p!{n\choose p}
\]
The number of different ways to classify $n_i$ elements in $i$ groups, when
the total number of elements is $N$, is
\[
\frac{N!}{\prod\limits_i n_i!}
\]
The probability $P(A)$ that an event $A$ occurs is defined by:
\[
P(A)=\frac{n(A)}{n(U)}
\]
where $n(A)$ is the number of events when $A$ occurs and $n(U)$ the total
number of events.
The probability $P(\neg A)$ that $A$ does not occur is: $P(\neg A)=1-P(A)$.
The probability $P(A\cup B)$ that $A$ and $B$ both occur is given by:
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. If $A$ and $B$ are independent, than holds:
$P(A\cap B)=P(A)\cdot P(B)$.
The probability $P(A|B)$ that $A$ occurs, given the fact that $B$ occurs, is:
\[
P(A|B)=\frac{P(A\cap B)}{P(B)}
\]
The average or mean value $\langle x\rangle$ of a collection of values is:
$\langle x\rangle=\sum_i x_i/n$. The standard deviation $\sigma_x$ in the
distribution of $x$ is given by:
\[
\sigma_x=\sqrt{\frac{\sum\limits_{i=1}^n(x_i-\langle x \rangle)^2}{n}}
\]
When samples are being used the sample variance $s$ is given by $\displaystyle s^2=\frac{n}{n-1}\sigma^2$.
The covariance $\sigma_{xy}$ of $x$ and $y$ is given by:
\[
\sigma_{xy}=\frac{\sum\limits_{i=1}^n(x_i-\langle x \rangle)(y_i-\langle y\rangle)}{n-1}
\]
The correlation coefficient $r_{xy}$ of $x$ and $y$ than becomes:
$r_{xy}=\sigma_{xy}/\sigma_x\sigma_y$.
The standard deviation in a variable $f(x,y)$ resulting from errors in
$x$ and $y$ is:
\[
\sigma^2_{f(x,y)}=\left(\frac{\partial f}{\partial x}\sigma_x\right)^2+\left(\frac{\partial f}{\partial y}\sigma_y\right)^2+
\frac{\partial f}{\partial x}\frac{\partial f}{\partial y}\sigma_{xy}
\]
- The Binomial distribution is the distribution describing a
sampling with replacement. The probability for success is $p$. The
probability $P$ for $k$ successes in $n$ trials is then given by:
\[
P(x=k)={n\choose k}p^k(1-p)^{n-k}
\]
The standard deviation is given by $\sigma_x=\sqrt{np(1-p)}$ and the
expectation value is $\varepsilon=np$.
- The Hypergeometric distribution is the distribution describing
a sampling without replacement in which the order is irrelevant.
The probability for $k$ successes in a trial with $A$ possible
successes and $B$ possible failures is then given by:
\[
P(x=k)=\frac{\displaystyle{A\choose k}{B\choose n-k}}{\displaystyle{A+B\choose n}}
\]
The expectation value is given by $\varepsilon=nA/(A+B)$.
- The Poisson distribution is a limiting case of the binomial
distribution when $p\rightarrow0$, $n\rightarrow\infty$ and also
$np=\lambda$ is constant.
\[
P(x)=\frac{\lambda^x {\rm e}^{-\lambda}}{x!}
\]
This distribution is normalized to $\displaystyle\sum\limits_{x=0}^\infty P(x)=1$.
- The Normal distribution is a limiting case of the binomial distribution
for continuous variables:
\[
P(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\langle x\rangle}{\sigma}\right)^2\right)
\]
- The Uniform distribution occurs when a random number $x$ is taken
from the set $a\leq x\leq b$ and is given by:
\[
\left\{\begin{array}{l}\displaystyle
P(x)=\frac{1}{b-a}~~~\mbox{if}~~~a\leq x\leq b\\
\\
P(x)=0~~~\mbox{in all other cases}
\end{array}\right.
\]
$\langle x\rangle=\frac{1}{2}(b-a)$ and $\displaystyle\sigma^2=\frac{(b-a)^2}{12}$.
- The Gamma distribution is given by:
\[
\left\{\begin{array}{l}\displaystyle
P(x)=\frac{x^{\alpha-1}{\rm e}^{-x/\beta}}{\beta^\alpha\Gamma(\alpha)}~~~\mbox{if}~~~0\leq y\leq\infty
\end{array}\right.
\]
with $\alpha>0$ and $\beta>0$. The distribution has the following
properties: $\langle x\rangle=\alpha\beta$, $\sigma^2=\alpha\beta^2$.
- The Beta distribution is given by:
\[
\left\{\begin{array}{l}\displaystyle
P(x)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\beta(\alpha,\beta)}~~~\mbox{if}~~~0\leq x\leq1\\
\\
P(x)=0~~~\mbox{everywhere else}
\end{array}\right.
\]
and has the following properties: $\displaystyle\langle x\rangle=\frac{\alpha}{\alpha+\beta}$,
$\displaystyle\sigma^2=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$.
For $P(\chi^2)$ holds: $\alpha=V/2$ and $\beta=2$.
- The Weibull distribution is given by:
\[
\left\{\begin{array}{l}\displaystyle
P(x)=\frac{\alpha}{\beta}x^{\alpha-1}{\rm e}^{-x^\alpha}~~~\mbox{if}~~~0\leq x\leq\infty\wedge\alpha\wedge\beta>0\\
\\
P(x)=0~~~\mbox{in all other cases}
\end{array}\right.
\]
The average is $\langle x\rangle=\beta^{1/\alpha}\Gamma((\alpha+1)\alpha)$
- For a two-dimensional distribution holds:
\[
P_1(x_1)=\int P(x_1,x_2)dx_2~~,~~~P_2(x_2)=\int P(x_1,x_2)dx_1
\]
with
\[
\varepsilon(g(x_1,x_2))=\iint g(x_1,x_2)P(x_1,x_2)dx_1dx_2=\sum_{x_1}\sum_{x_2}g\cdot P
\]
When there exists a relation between the quantities $x$ and $y$ of the form
$y=ax+b$ and there is a measured set $x_i$ with related $y_i$, the following
relation holds for $a$ and $b$ with $\vec{x}=(x_1,x_2,...,x_n)$ and
$\vec{e}=(1,1,...,1)$:
\[
\vec{y}-a\vec{x}-b\vec{e}\in<\vec{x},\vec{e}>^\perp
\]
From this follows that the inner products are 0:
\[
\left\{
\begin{array}{l}
(\vec{y},\vec{x})-a(\vec{x},\vec{x})-b(\vec{e},\vec{x})=0\\
(\vec{y},\vec{e})-a(\vec{x},\vec{e})-b(\vec{e},\vec{e})=0
\end{array}\right.
\]
with $(\vec{x},\vec{x})=\sum\limits_i x_i^2$, $(\vec{x},\vec{y})=\sum\limits_ix_iy_i$,
$(\vec{x},\vec{e})=\sum\limits_ix_i$ and $(\vec{e},\vec{e})=n$.
$a$ and $b$ follow from this.
A similar method works for higher order polynomial fits: for a second order
fit holds:
\[
\vec{y}-a\vec{x^2}-b\vec{x}-c\vec{e}\in<\vec{x^2},\vec{x},\vec{e}>^\perp
\]
with $\vec{x^2}=(x_1^2,...,x_n^2)$.
The correlation coefficient $r$ is a measure for the quality of a fit.
In case of linear regression it is given by:
\[
r=\frac{n\sum xy-\sum x\sum y}{\sqrt{(n\sum x^2-(\sum x)^2)(n\sum y^2-(\sum y)^2)}}
\]