Probability distribution

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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

<math>

F(x) = {\rm Pr} \left[ X \le x \right] </math>

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

<math>

{\rm Pr} \left[ a \le X \le b \right] = \int_a^b f(x)\,dx </math>

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 in theory or applications that they have been given specific names:

See also

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

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