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.

Related Topics:
Mathematics - Interval - Real number - Probability - Probability axioms - Probability measure - Borel algebra

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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.

Related Topics:
Probability measure - Kolmogorov axioms - Measurable space

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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 the probability Pr, i.e. the probability that the variable X will take a value in the interval .

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The probability distribution of the variable X can be uniquely described by its cumulative distribution function F(x), which is defined by

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: F(x) = Prleft

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for any x in R.

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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 = 0 for all x in R.

Related Topics:
Discrete random variable - Countable - Continuous

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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

Related Topics:
Probability density function - Lebesgue integrable

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:

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Pr left = int_a^b f(x),dx

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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.

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• The support of a distribution is the smallest closed set whose complement has probability zero.
• The probability distribution of the sum of two 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.
 
Probability distribution: List of important probability distributions ►