14.2 Some Math Notation

Variables

A set of values, labeled with subscripts...

$$\begin{array}{c}{x}_1=1\\ x_2=5\\ x_3=-3\\ x_4=2\\ x_5=8\end{array}$$

referred collectively as $x_i$.

Sums

$$\begin{array}{c}{x}_1+x_2+x_3+x_4+x_5=1+5+(-3)+2+8=13\end{array}$$

is equivalent to

$$\begin{array}{c}{\sum }_{i=1}^5{x}_i=1+5+(-3)+2+8=13\end{array}$$

Products

$$\begin{array}{c}{x}_1·x_2·x_3·x_4·x_5=1·5·(-3)·2·8=-240\end{array}$$

is equivalent to

$$\begin{array}{c}{\prod }_{i=1}^5{x}_i=1·5·(-3)·2·8=-240\end{array}$$

Sample Mean

The sample mean of a set of numbers is defined as...

$$\begin{array}{c}\overline{x}\equiv \frac{x_1+x_2+\cdots x_N}{N}\end{array}$$

In the example above

$$\begin{array}{c}\overline{x}\equiv \frac{x_1+x_2+x_3+x_4+x_5}{5}=2\frac{3}{5}\end{array}$$

It can also be written

$$\begin{array}{c}\overline{x}\equiv \frac{{\sum }_{i=1}^N{x}_i}{N}\end{array}$$

or

$$\begin{array}{c}\overline{x}\equiv \frac{{\sum }_i{x}_i}{N}\end{array}$$

Sample Standard Deviation

$$\begin{array}{c}{s}^2\equiv \frac{1}{N-1}{\sum }_{i=1}^N(x-\overline{x}{)}^2\end{array}$$

$$\begin{array}{c}s\equiv \sqrt{\frac{1}{N-1}{\sum }_{i=1}^N(x-\overline{x}{)}^2}\end{array}$$

Estimates

Any specific estimate of a parameter, such as $\theta$, is denoted with a hat, such as $\widehat{\theta }$.

Factorials

Factorials are defined as

$$\begin{array}{c}N!=1·2·3\cdots (N-1)·N\end{array}$$

for example

$$\begin{array}{c}5!=1·2·3·4·5=120\end{array}$$

The N-choose-k notation is a shorthand for the factorials that arise in binomial and Beta distributions.

$$\begin{array}{c}\left(\genfrac{}{}{0ex}{}{N}{k}\right)\equiv \frac{N!}{k!(N-k)!}\end{array}$$

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.