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2. Probability and statistics

2.1 Combinations

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

$\frac{N!}{\prod\limits_i n_i!}$

2.2 Probability theory

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: $P(A|B)=\frac{P(A\cap B)}{P(B)}$

2.3 Statistics

2.3.1 General

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: $\sigma^2_{f(x,y)}=\left(\frac{\partial f}{\partial x}\sigma_x\right)^2+\left(\frac{\partial f}{\partial y}\sigma_y\right)^2+ \frac{\partial f}{\partial x}\frac{\partial f}{\partial y}\sigma_{xy}$

2.3.2 Distributions

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 $\varepsilon(g(x_1,x_2))=\iint g(x_1,x_2)P(x_1,x_2)dx_1dx_2=\sum_{x_1}\sum_{x_2}g\cdot P$

2.4 Regression analyses

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: $r=\frac{n\sum xy-\sum x\sum y}{\sqrt{(n\sum x^2-(\sum x)^2)(n\sum y^2-(\sum y)^2)}}$