In mathematics and statistics, a probability distribution describes how probabilities are distributed on events. It assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. Probability distributions are special cases of probability measures defined over the Borel algebra on the reals.

Formal definition

Every random variable has a probability distribution that characterizes it completely. If X is a random variable, the corresponding probability distribution assigns to the interval [a, b] the probability Pr[a ≤ X ≤ b], i.e. the probability that the variable X will take a value in the interval [a, b]. The probability distribution of the variable X can be uniquely described by its cumulative distribution function F(x), which is defined by

for any x in R.

A distribution is called discrete if its cumulative distribution function only increases in jumps, or equivalently that it belongs to a discrete random variable, a random variable which is fully characterised by the probabilities it assigns to a certain finite or countable set of values. By one convention, a distribution is called continuous if its cumulative distribution function is continuous, which means that it belongs to a random variable X for which Pr[ X = x ] = 0 for all x in R. Another convention reserves the term continuous probability distribution for absolutely continuous distributions. These can be expressed by a probability density function: a non-negative Lebesgue integrable function f defined on the real numbers such that

for all a and b. Of course, discrete distributions do not admit such a density; there also exist some continuous distributions like the devil's staircase that do not admit a density.

Here is called the probability mass function.

• The support of a distribution is the smallest closed set whose complement has probability zero.
• The probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
• The probability distribution of the difference of two random variables is the cross-correlation of each of their distributions.

List of important probability distributions

Several probability distributions are so important in theory or applications that they have been given specific names:

Discrete distributions

With finite support

• The Bernoulli distribution, which takes value 1 with probability p and value 0 with probability q = 1 − p.
• The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2.
• The binomial distribution describes the number of successes in a series of independent Yes/No experiments.
• The degenerate distribution at x₀, where X is certain to take the value x₀. This does not look random, but it satisfies the definition of random variable because although its output is determinate, its input is random. This is useful because it puts deterministic variables and random variables in the same formalism.
• The discrete uniform distribution, where all elements of a finite set are equally likely. This is supposed to be the distribution of a balanced coin, an unbiased die, a casino roulette or a well-shuffled deck. Also, one can use measurements of quantum states to generate uniform random variables. All these are "physical" or "mechanical" devices, subject to design flaws or perturbations, so the uniform distribution is only an approximation of their behaviour. In digital computers, pseudo-random number generators are used to produce a statistically random discrete uniform distribution.
• The hypergeometric distribution, which describes the number of successes in the first m of a series of n Yes/No experiments, if the total number of successes is known.
• Zipf's law or the Zipf distribution. A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language.
• The Zipf-Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution.

With infinite support

• The Boltzmann distribution, a discrete distribution important in statistical physics which describes the probabilities of the various discrete energy levels of a system in thermal equilibrium. It has a continuous analogue. Special cases include:
  • The Gibbs distribution
  • The Maxwell-Boltzmann distribution
  • The Bose-Einstein distribution
  • The Fermi-Dirac distribution
• The geometric distribution, a discrete distribution which describes the number of attempts needed to get the first success in a series of independent Yes/No experiments.

• The logarithmic (series) distribution
• The negative binomial distribution, a generalization of the geometric distribution to the nth success.
• The parabolic fractal distribution
• The Poisson distribution, which describes a very large number of individually unlikely events that happen in a certain time interval.

• The Skellam distribution, the distribution of the difference between two independent Poisson-distributed random variables.
• The Yule-Simon distribution
• The zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists. It is the Zipf distribution for an infinite number of elements.