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The geometric distribution describes the waiting time until the first success in a sequence of independent Bernoulli trials. One version counts the failures before the first success, and the other counts the trials up to and including it. The node geometric distribution gives the precise definition.

The normal distribution with mean $\mu$ and variance $\sigma^2$ is the continuous distribution on $\mathbb{R}$ with the bell-shaped density $\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)}$. By the central limit theorem, it is the approximate law of a sum of many small independent effects. The node normal distribution gives the precise definition.

A relation is symmetric when it always holds in both directions.

An analytic function is a function that is given locally by a convergent power series. Analytic functions are smooth, but not all smooth functions are analytic. The node analytic function gives the precise definition.

A topological space is metrizable when its topology comes from some metric. Metrization theorems give conditions on the topology alone that make a space metrizable.

A central moment of a random variable is a moment of its deviation from the mean. The second central moment is the variance. The node central moment gives the precise definition.

A function is injective when different inputs always give different values.

Student's t-distribution is a continuous law on $\mathbb{R}$ that is symmetric about $0$ and bell-shaped, like the standard normal law, but with heavier tails. It is the law of the standardized sample mean when one estimates the standard deviation from the data. The node Student's t-distribution gives the precise definition.

The chi-squared distribution with $k$ degrees of freedom is the law of a sum of the squares of $k$ independent standard normal random variables. It is central in tests and confidence intervals for variances. The node chi-squared distribution gives the precise definition.

A step function is a function that is constant on each piece of a partition of an interval into finitely many pieces. Its integral is a finite sum of areas of rectangles. The node step function gives the precise definition.

The continuous uniform distribution on an interval $[a, b]$ spreads the probability evenly over the interval: the probability of a subinterval is proportional to its length. The node continuous uniform distribution gives the precise definition.

The gamma distribution is a two-parameter family of continuous distributions on $(0, \infty)$. It contains the exponential, Erlang and chi-squared distributions. The node gamma distribution gives the precise definition.

A complete metric space is a metric space in which each Cauchy sequence converges to a point of the space. The real numbers are complete, and the rational numbers are not.

A relation is well-founded when each nonempty subset has a minimal element for it. Then there is no infinite descending chain.

A probability distribution describes how probability is spread over the possible values of a random quantity. Formally it is a probability measure on a space of values, such as the real line with its Borel sets. The node probability distribution gives the precise definition.

The exponential distribution with rate $\lambda$ is the law of a waiting time with no memory. An example is the time between two events of a Poisson process. The node exponential distribution gives the precise definition.

The beta distribution is a two-parameter family of continuous distributions on the interval $[0, 1]$. It is often used as a distribution for an unknown probability. The node beta distribution gives the precise definition.

An equivalence relation is a relation that behaves like equality: it is reflexive, symmetric and transitive.

A function is bijective when it is both injective and surjective, so each element of the codomain is the value at exactly one input.

A relation is transitive when it passes along chains. If a first element is related to a second, and the second to a third, then the first is related to the third.

The Cauchy distribution is a continuous distribution on $\mathbb{R}$ with a bell-shaped density and heavy tails. It has no mean and no variance. The node Cauchy distribution gives the precise definition.

A conjugate of a subgroup $H$ of a group $G$ is a subgroup $gHg^{-1}$ for some $g \in G$. Conjugate subgroups are isomorphic.

Each function induces an equivalence relation on its domain: two elements are equivalent when the function has the same value at both.

The free group on a set $S$ is the group of reduced words on $S$, with concatenation followed by reduction as the operation. It is the group generated by $S$ with no relations other than those that the group axioms force.

A relation is antisymmetric when two different elements are never related in both directions.

The identity function of a set sends each element to itself.

A Boolean ring is a ring in which every element is idempotent. Boolean rings are Boolean algebras written in the language of rings, with exclusive or as addition. The node Boolean ring gives the precise definition.

A proper metric space is a metric space in which each closed ball is compact. In such a space, a set is compact if and only if it is closed and bounded.

A smooth function is a function with continuous derivatives of all orders. Smoothness is the top of the scale of differentiability classes $C^k$. The node smooth function gives the precise definition.

A metric space is a set with a distance function between its points. The distance is zero only from a point to itself, it is symmetric, and it satisfies the triangle inequality. Metric spaces are the setting for limits, continuity and completeness beyond the real line.

An uncountable set is a set that is not countable. It is too large to be listed in a sequence.

The binomial distribution with parameters $n$ and $p$ is the distribution of the number of successes in $n$ independent Bernoulli trials. Each trial has the success probability $p$. The node binomial distribution gives the precise definition.

A discrete random variable is a random variable whose values lie, with probability one, in a countable set. Its law is described by the probabilities of the single values. The node discrete random variable gives the precise definition.

The hypergeometric distribution describes the number of successes in a sample drawn without replacement from a finite population. The node hypergeometric distribution gives the precise definition.

The elementary symmetric polynomials in $n$ variables are the sums of all products of $k$ distinct variables, for $k = 0, \dots, n$. They generate all symmetric polynomials. The definition of the elementary symmetric polynomials gives the formula.

A relation on a set is reflexive when each element is related to itself.

The Poisson distribution with parameter $\lambda$ describes the number of events in a fixed interval of time or space. It applies when the events occur independently at a constant mean rate. The node Poisson distribution gives the precise definition.

The negative binomial distribution describes the number of failures before the $r$-th success in a sequence of independent Bernoulli trials. The node negative binomial distribution gives the precise definition.

A composition series of a group is a subnormal series in which each factor group is simple. It cannot be refined further, and its factors are unique up to order by the Jordan–Hölder theorem.

A simple group is a nontrivial group whose only normal subgroups are the trivial subgroup and the group itself. The finite simple groups are the building blocks of all finite groups.