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# Probability distribution

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.

More generally, a probability distribution can be assigned to any measurable space. See Kolmogorov axioms.

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

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

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.

## List of important probability distributions

Several probability distributions are so important that they have been given specific names:

• Discrete distributions
• Continuous distributions
• Supported on a finite interval
• The uniform distribution on [a,b], where all points in a finite interval are equally likely.
• 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 exponential distribution, which describes the time between rare random events.
• The Gamma distribution, which describes the time until n rare random events occur.
• The Log-normal distribution, describing variables which can be modelled as the product of many small independent positive variables.
• The Weibull distribution, of which the exponential distribution is a special case, is used to model the lifetime of technical devices.
• The chi-square distribution, which is the sum of the squares of n independent Gaussian random variables. It is a special case of the Gamma distribution, and it is used in goodness-of-fit tests in statistics.
• Supported on the whole real line
• The normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the central limit theorem: every variable that can be modelled as a sum of many small independent variables is approximately normal.
• Student's t-distribution, useful for estimating unknown means of Gaussian populations.
• The Cauchy distribution, an example of a distribution which does not have an expected value or a variance. In physics it is usually called a Lorentzian, and it is the distribution of the energy of an unstable state in quantum mechanics. In particle physics, the extremely short-lived particles associated to unstable states are called resonances.
• Joint distributions
• Two or more random variables on the same sample space
• Bivariate distribution
• Conditional distribution
• Multivariate distribution
• Multinomial distribution, a generalization of the binomial distribution.

list of statistical topics -- random variable -- cumulative distribution function -- probability density function -- likelihood

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