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  is the measure  on  defined by setting

where  is a probability space,  is a measurable space, and  a measure on  with . 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.

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.