Mathematical Notation Glossary

The Greek letter mu, commonly used to represent the population mean or expected value of a distribution.

The Greek letter sigma, used for standard deviation (σ) or variance (σ²). Measures the spread of a distribution around its mean.

Capital sigma indicates summation - adding up a series of terms. The subscript and superscript specify the range.

The Greek letter alpha, often used for the tail exponent in power law distributions, or as a general parameter.

The Greek letter beta, commonly used as a parameter in distributions or regression coefficients.

The Greek letter gamma, often used to represent skewness (third standardized moment) or as a parameter.

The Greek letter kappa, sometimes used for kurtosis (fourth standardized moment).

Capital omega represents the sample space - the set of all possible outcomes of a random experiment.

The Greek letter delta, commonly used to represent a small change or difference in a quantity.

Capital gamma denotes the gamma function Γ(n), which extends factorial to non-integers: Γ(n) = (n-1)! for positive integers.

The Greek letter xi, commonly used as the shape parameter in Extreme Value Theory (GEV distribution). Determines tail behavior.

The Greek letter nu, commonly used for degrees of freedom in t-distributions and chi-squared distributions.

The Greek letter lambda, often used as a rate parameter in Poisson or exponential distributions.

The mathematical constant pi, the ratio of a circle's circumference to its diameter. Appears frequently in probability density functions.

The infinity symbol represents an unbounded quantity. Often appears in integration limits.

Indicates membership in a set. "x ∈ A" means "x is an element of set A" or "x is in A".

Universal quantifier. "∀x" means "for all x" or "for every x".

Existential quantifier. "∃x" means "there exists an x" or "for some x".

The empty set, a set containing no elements. Also written as {}.

Indicates that one set is contained within another. "A ⊂ B" means "A is a subset of B".

Set union. "A ∪ B" is the set of elements in A or B (or both).

Set intersection. "A ∩ B" is the set of elements in both A and B.

The set of all real numbers, including rational and irrational numbers. Continuous random variables typically take values in ℝ or subsets of ℝ.

The integral sign indicates integration - a continuous analog of summation. Used extensively for computing probabilities and expectations.

The limit of a function or sequence. "lim_{x→a} f(x)" is the value f(x) approaches as x gets closer to a.

Indicates a partial derivative - the derivative with respect to one variable while holding others constant.

The probability function. P(A) gives the probability of event A occurring.

The expectation operator. E[X] is the expected value (mean) of random variable X.

The variance of a random variable, measuring its spread around the mean. Var(X) = E[(X - μ)²].

Indicates the distribution of a random variable. "X ~ N(0,1)" means "X is distributed as standard normal".

The probability density function (for continuous distributions). Describes the relative likelihood of different values.

The cumulative distribution function. F(x) = P(X ≤ x) gives the probability of being at or below x.

The survival function (also called tail function). S(x) = P(X > x) = 1 - F(x) gives the probability of exceeding x.

The covariance measures the joint variability of two random variables. Cov(X,Y) = E[(X - μX)(Y - μY)].

Independent and identically distributed: random variables that are mutually independent and share the same probability distribution.

Almost surely (or almost certain) means an event happens with probability 1. Used in convergence theorems.

The independence symbol indicates that random variables or events are statistically independent.

f(x) ~ g(x) means lim f(x)/g(x) = 1 as x → ∞. The functions become indistinguishable in ratio for large x. Used extensively to describe tail behavior.

f(x) = O(g(x)) means |f(x)| ≤ C|g(x)| for some constant C and sufficiently large x. It provides an upper bound on the growth rate.

f(x) = o(g(x)) means f(x)/g(x) → 0 as x → ∞. The function f becomes negligible compared to g. Stronger than big-O.

A function L(x) is slowly varying if L(tx)/L(x) → 1 for all t > 0. Examples include constants, ln(x), and (ln x)^k. Used in fat tail theory to capture deviations from pure power laws.

A function f(x) is regularly varying with index α if f(tx)/f(x) → t^α for all t > 0. Can be written as f(x) = x^α L(x) where L is slowly varying. Characterizes fat-tailed distributions.

A distribution F is in the domain of attraction of G (written F ∈ D(G)) if the maximum of n i.i.d. samples from F, properly normalized, converges to G. Determines which extreme value type applies.