# Random Variable

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## random variable

[′ran·dəm ′ver·ē·ə·bəl]## 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

*x _{1}, x_{2}, …, x_{n}, …*

and the probabilities associated with these values

*p _{1,}, p_{2}, …, 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 D*X*. Such characteristics as the median, mode, and quantile are less often used. (*See also*PROBABILITY THEORY.)

### REFERENCES

Gnedenko, B. V.*Kurs teorii veroiatnostei*, 5th ed. Moscow, 1969.

Cramer, H.

*Sluchainye velichiny i raspredeleniia veroiatnostei.*Moscow, 1947. (Translated from English.)