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, StatisticsExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Statistical DistributionCite this as:
Weisstein, Eric W. "Statistical Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StatisticalDistribution.html