Random Variable (redirected from Random mapping)

random variable

In statistics, a function that can take on either a finite number of values, each with an associated probability, or an infinite number of values, whose probabilities are summarized by a density function. Used in studying chance events, it is defined so as to account for all possible outcomes of the event. When these are finite (e.g., the number of heads in a three-coin toss), the random variable is called discrete and the probabilities of the outcomes sum to 1. If the possible outcomes are infinite (e.g., the life expectancy of a light bulb), the random variable is called continuous and corresponds to a density function whose integral over the entire range of outcomes equals 1. Probabilities for specific outcomes are determined by summing probabilities (in the discrete case) or by integrating the density function over an interval corresponding to that outcome (in the continuous case).

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random variable [′ran·dəm ′ver·ē·ə·bəl]

(mathematics)

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.

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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, …, x_n, …

and the probabilities associated with these values

p_1,, p₂, …, p_n, …

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 P_x(a, b) of the inequality a ≤ x < b. Random variables are encountered particularly often for which there exists a function P_x(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.)

REFERENCES

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