A rule that assigns a real number to each outcome of a random experiment is called a random variable. It is governed by a function of the variable. A random variable is usually denoted by any of the capital Latin letters X, Y, Z…….
Let S be the sample space associated with a given random experiment. A real – valued function defined on S and taking values in R(-∞,∞) is called a one – dimensional random variable. If the function values are ordered pairs of real numbers the function is said to be a two – dimensional random variable. More generally, an n – dimensional random variable is simply a function whose domain is S and whose range is a collection of n – tuples of real numbers.
For a mathematical and rigorous definition of the random variable, let us consider the probability space, the triplet (S, B, P), where S is the sample space, the space of outcomes, B is the σ – field of subsets in S, and P is a probability function on B.
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