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Statistical Distribution


The distribution of a variable is a description of the relative numbers of times each possible outcome will occur in a number of trials. The function describing the probability that a given value will occur is called the probability density function (abbreviated PDF), and the function describing the cumulative probability that a given value or any value smaller than it will occur is called the distribution function (or cumulative distribution function, abbreviated CDF).

Formally, a distribution can be defined as a normalized measure, and the distribution of a random variable x is the measure P_x on S^' defined by setting

 P_x(A^')=P{s in S:x(s) in A^'},

where (S,S,P) is a probability space, (S,S) is a measurable space, and P a measure on S with P(S)=1. If the measure is a Radon measure (which is usually the case), then the statistical distribution is a generalized function in the sense of a generalized function.


See also

Continuous Distribution, Discrete Distribution, Distribution Function, Generalized Function, Measurable Space, Measure, Probability, Probability Density Function, Random Variable, Statistics

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References

Doob, J. L. "The Development of Rigor in Mathematical Probability (1900-1950)." Amer. Math. Monthly 103, 586-595, 1996.Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, 2000.

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Statistical Distribution

Cite this as:

Weisstein, Eric W. "Statistical Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StatisticalDistribution.html

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