14 Notation
14.1 Random variables, vectors, and matrices
- Random variables are capital letters such as X and Y.
- A particular value of X is x.
- Vectors are bold lowercase symbols (e.g., \boldsymbol{a}, \boldsymbol{\theta}) or italic lowercase with arrows (e.g., \vec{a}, \vec{\theta}).
- Matrices are bold uppercase symbols (e.g., \boldsymbol{A}, \boldsymbol{\Theta}).
| Expression | Meaning |
|---|---|
| \boldsymbol{A'} | The transpose of matrix \boldsymbol{A} |
| \boldsymbol{A}^{-1} | The inverse of matrix \boldsymbol{A} |
| \mathtt{diag}(\boldsymbol{A}) | The vector formed from the diagonal of matrix \boldsymbol{A} |
| \mathtt{diag}(\boldsymbol{a}) | The diagonal matrix formed from vector \boldsymbol{a} |
| \boldsymbol{1}_k | A column vector of k ones. If k is absent, the length of \boldsymbol{1} can be inferred by context. |
| \boldsymbol{I}_k | A k \times k identity matrix. If k is absent, the dimensions of \boldsymbol{I} can be inferred by context. |
14.2 Sets and intervals
A set is a collection of distinct objects.
| Expression | Meaning |
|---|---|
| \{a,b\} | A set consisting of two numbers, a and b |
| \lbrack a,b\rbrack | The set of real numbers in the interval between a and b, inclusive |
| (a,b) | The set of real numbers in the interval between a and b, exclusive |
| (a,b\rbrack | The set of real numbers in the interval between a and b, excluding a but including b |
| \mathbb{R} | The set of all real numbers [-\infty,\infty] |
| \mathbb{Z} | The set of all integers \{\ldots,-2,-1,0,1,2,\ldots\} |
| \mathbb{N}_0 | The set of all non-negative integers: \{ 0,1,2,...\} |
| \mathbb{N}_1 | The set of all positive integers: \{ 1,2,3,...\} |
| \in | Is a member of (i.e., X \in (a,b) means that X is a member of the set of numbers between a and b) |
14.3 Summation
There are several ways to show that a variable is to be summed. The summation sign looks like a really big Σ, the capital Greek letter sigma. If \boldsymbol{x} is a vector with k elements, the sum of all k in \boldsymbol{x} is:
\sum_{i = 1}^k x_i
With matrix algebra,
\sum_{i = 1}^k x_i=\boldsymbol{1}'\boldsymbol{x}
14.4 Statistics
| Expression | Meaning |
|---|---|
| \mu_X, m_X | The population and sample mean of X |
| \sigma_X, s_X | The population and sample standard deviation of X |
| \sigma_X^2, s_X^2 | The population and sample variance of X |
| \gamma_1, g_1 | The population and sample skewness |
| \gamma_2, g_2 | The population and sample kurtosis |
14.5 Other
| Expression | Meaning |
|---|---|
| \binom{n}{k} | The binomial coefficient. It is just a shortcut notation for \binom{n}{k}=\frac{n!}{k!\left(n-k\right)!}. Read aloud, \binom{n}{k} is ``n choose k’’ or the number of combinations that n things have when taken k at a time. |
| f_X(x;\boldsymbol{\theta}) | The probability density function or probability mass function of X with parameters \boldsymbol{\theta} |
| F_X(x;\boldsymbol{\theta}) | The cumulative distribution function of X with parameters \boldsymbol{\theta} |
| \mathcal{E}\left(X\right) | The expected value of X |