Random Variable

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random variable

[′ran·dəm ′ver·ē·ə·bəl]
A measurable function on a probability space; usually real valued, but possibly with values in a general measurable space. Also known as chance variable; stochastic variable; variate.

Random Variable


in probability theory, a quantity whose assumption of particular values is controlled by chance. Each of the values the random variable can assume has a certain probability. Thus, the number of spots on the top face of a die is a random variable that assumes the values 1, 2, 3,4, 5 and 6; the probability of each value is 1/6.

If a random variable X has a finite or infinite sequence of distinct values, the probability distribution function (distribution law) of X can be specified by indicating these values

x1, x2, …, xn, …

and the probabilities associated with these values

p1,, p2, …, pn, …

This type of random variable is said to be discrete. In other cases, the probability distribution function can be specified by indicating for each closed interval Δ = [a, b] the probability Px(a, b) of the inequality ax < b. Random variables are encountered particularly often for which there exists a function Px(x), called the probability density function, such that

This type of random variable is said to be continuous.

A number of general properties of the probability distribution function of a random variable can be described sufficiently fully by a small set of numerical characteristics. The most frequently used characteristics are the mathematical expectation EX of the random variable X and the variable’s variance DX. Such characteristics as the median, mode, and quantile are less often used. (See alsoPROBABILITY THEORY.)


Gnedenko, B. V. Kurs teorii veroiatnostei, 5th ed. Moscow, 1969.
Cramer, H. Sluchainye velichiny i raspredeleniia veroiatnostei. Moscow, 1947. (Translated from English.)
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