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# Probability distribution

In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra on the reals.

A probability distribution is a special case of the more general notion of a probability measure, which is a function that assigns probabilities satisfying the Kolmogorov axioms to the measurable sets of a measurable space.

Every random variable gives rise to a probability distribution, and this distribution contains most of the important information about the variable. If X is a random variable, the corresponding probability distribution assigns to the interval [a, b] the probability Pr[aXb], 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

$F(x) = \Pr\left[ X \le x \right]$

for any x in R.

A distribution is called discrete if its cumulative distribution function consists of a sequence of finite jumps, which means that it belongs to a discrete random variable X: a variable which can only attain values from a certain finite or countable set. 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.

The so-called absolutely continuous distributions can be expressed by a probability density function: a non-negative Lebesgue integrable function f defined on the reals such that

$\Pr \left[ a \le X \le b \right] = \int_a^b f(x)\,dx$

for all a and b. That discrete distributions do not admit such a density is unsurprising, but there are continuous distributions like the devil's staircase that also do not admit a density.

The support of a distribution is the smallest closed set whose complement has probability zero.

 Contents

## 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, which 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. 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 produced a statistically random discrete uniform distribution.
• The Ewens sampling formula is a probability distribution on the set of all partitions of an integer n, arising in population genetics.
• The hypergeometric distribution, which describes the number of successes in the first m of a series of n independent 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.

### Continuous distributions

#### Supported on a bounded interval

• The Beta distribution on [0,1], of which the uniform distribution is a special case, and which is useful in estimating success probabilities.

#### Supported on semi-infinite intervals, usually [0,∞)

• The Gamma distribution, which describes the time until n consecutive rare random events occur in a process with no memory.
• Fisher's z-distribution
• The half-normal distribution
• The log-logistic distribution
• The log-normal distribution, describing variables which can be modelled as the product of many small independent positive variables.
• The Pareto distribution, or "power law" distribution, used in the analysis of financial data and critical behavior.
• The Rice distribution
• The type-2 Gumbel distribution
• The Wald distribution
• The Weibull distribution, of which the exponential distribution is a special case, is used to model the lifetime of technical devices.