PSTAT 120B: Mathematical Statistics, I
Statistical Notation
I am sympathetic to the fact that, at times, the notation used in statistics can seem a bit overwhelming. As such, I’ve crafted this page to outline a few key notations and symbols used in Statistics - I encourage you to refer to this page often! (Of course, please feel free to ask me directly in Lecture or Office Hours if I use notation that is unfamiliar to you.)
Greek Alphabet
Table 1: The Greek Alphabet
| Uppercase | Lowercase | Name |
|---|---|---|
| A | \(\alpha\) | alpha |
| B | \(\beta\) | beta |
| \(\Gamma\) | \(\gamma\) | gamma |
| \(\Delta\) | \(\delta\) | delta |
| E | \(\varepsilon\) | epsilon |
| Z | \(\zeta\) | zeta |
| H | \(\eta\) | nu |
| \(\Theta\) | \(\theta\) / \(\vartheta\) | theta |
| I | \(\iota\) | iota |
| K | \(\kappa\) | kappa |
| \(\Lambda\) | \(\lambda\) | lambda |
| M | \(\mu\) | mu |
| Uppercase | Lowercase | Name |
|---|---|---|
| N | \(\nu\) | nu |
| \(\Xi\) | \(\xi\) | xi |
| O | \(\omicron\) | omicron |
| \(\Pi\) | \(\pi\) | pi |
| P | \(\rho\) | rho |
| \(\Sigma\) | \(\sigma\) / \(\varsigma\) | sigma |
| T | \(\tau\) | tau |
| Y | \(\upsilon\) | upsilon |
| \(\Phi\) | \(\phi\) | phi |
| X | \(\chi\) | chi |
| \(\Psi\) | \(\psi\) | psi |
| \(\Omega\) | \(\omega\) | omega |
Some Common Statistical Uses
- \(\mu\): often used to denote mean/expected value
- \(\sigma\): often used to denote standard deviations
- \(\theta\): often used as a placeholder for an arbitrary parameter
- \(\lambda, \ \rho\): often used to denote rates
- \(\nu\): often used to denote the degrees of freedom of a \(\chi^2\) distribution
- \(\Gamma(t)\): the Gamma function \[ \Gamma(r) := \int_{0}^{\infty} x^{r - 1} e^{-x} \ \mathrm{d}x \]
- \(\psi(t)\): the Digamma function: \[ \psi(t) := \frac{\mathrm{d}}{\mathrm{d}t} \ln[\Gamma(t)] = \frac{\Gamma'(t)}{\Gamma(t)} \]
- Sometimes, related parameters will be named after consecutive letters in the alphabet. For instance, we may denote the mean of a random variable \(X\) by \(\mu\) and the mean of another random variable \(Y\) by \(\nu\), and \(\mu\) and \(\nu\) appear consecutively in the Greek alphabet. Similarly for variances; \(\sigma^2\) and \(\tau^2\) are commonly used to denote variances.
Other Notations
In this class, I will often write \(X \sim f_X\) to mean “let \(X\) be a random variable with density function \(f_X(x)\)”, and \(X \sim F_X\) to mean “let \(X\) be a random variable with distribution function \(F_X(x)\).”
The symbol \(\forall\) is called the universal quantifier, and is essentially a mathematical way of writing “for all” or “for every”. As an example: \[ (\forall x \in \mathbb{R})[f_X(x) \geq 0] \] is read “for every real number \(x\), the value of \(f_X(x)\) will be nonnegative”.
The symbol \(\exists\) is called the existential quantifier, and is essentially a mathematical way of writing “there exists” or “for at least one”. As an example: \[ (\exists x \in \mathbb{R})[x^2 = 2] \] is read “there exists a real number \(x\) whose square is 2”.
I often use the symbol \(:=\) to mean “definitionally equal to”. For example, given a random variable \(Y\), I’ll often write \(U := Y^2\) to mean “let \(U\) be another random variable that is, by definition, equal to the square of \(Y\).”
I often use the symbol \(\stackrel{!}{=}\) to mean “set equal to”. For example, solving \(f'(x) \stackrel{!}{=} 0\) for \(x\) gives us the set of critical values for the function \(f(\cdot)\).