14.2 Some Math Notation

Variables

A set of values, labeled with subscripts...

$$\begin{eqnarray*} x_{1}&=&1 \\ x_{2}&=&5 \\ x_{3}&=&-3 \\ x_{4}&=&2 \\ x_{5}&=&8 \end{eqnarray*}$$

referred collectively as $x_{i}$.

Sums

$$\begin{eqnarray*} x_{1}+x_{2}+x_{3}+x_{4}+x_{5} = 1+5+(-3)+2+8 = 13 \end{eqnarray*}$$

is equivalent to

$$\begin{eqnarray*} \sum_{i=1}^{5} x_{i} = 1+5+(-3)+2+8 = 13 \end{eqnarray*}$$

Products

$$\begin{eqnarray*} x_{1}\cdot x_{2}\cdot x_{3}\cdot x_{4}\cdot x_{5} = 1\cdot 5\cdot (-3)\cdot 2\cdot 8 = -240 \end{eqnarray*}$$

is equivalent to

$$\begin{eqnarray*} \prod_{i=1}^{5} x_{i} = 1\cdot 5\cdot (-3)\cdot 2\cdot 8 = -240 \end{eqnarray*}$$

Sample Mean

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

$$\begin{eqnarray*} \bar{x} \equiv \frac{x_{1}+x_{2}+\cdots x_{N}}{N} \end{eqnarray*}$$

In the example above

$$\begin{eqnarray*} \bar{x} \equiv \frac{x_{1}+x_{2}+x_{3}+x_{4}+x_{5}}{5} = 2\frac{3}{5} \end{eqnarray*}$$

It can also be written

$$\begin{eqnarray*} \bar{x} \equiv \frac{\sum_{i=1}^{N} x_{i}}{N} \end{eqnarray*}$$

or

$$\begin{eqnarray*} \bar{x} \equiv \frac{\sum_{i} x_{i}}{N} \end{eqnarray*}$$

Sample Standard Deviation

$$\begin{eqnarray*} s^{2}\equiv \frac{1}{N-1} \sum_{i=1}^{N} (x-\bar{x})^{2} \end{eqnarray*}$$

$$\begin{eqnarray*} s\equiv \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} (x-\bar{x})^{2}} \end{eqnarray*}$$

Estimates

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

Factorials

Factorials are defined as

$$\begin{eqnarray*} N! = 1 \cdot 2 \cdot 3 \cdots (N-1) \cdot N \end{eqnarray*}$$

for example

$$\begin{eqnarray*} 5! = 1 \cdot 2 \cdot 3\cdot 4 \cdot 5 = 120 \end{eqnarray*}$$

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

$$\begin{eqnarray*} \binom{N}{k} \equiv \frac{N!}{k!(N-k)!} \end{eqnarray*}$$

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.