Random variable: Difference between revisions

Revision as of 09:03, 19 April 2002

Revision as of 11:34, 19 April 2002

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We can think of a '''random variable''' as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a [[random]] result. For example, rolling a die and recording the outcome yields a random variable with range {1,2,3,4,5,6}. Picking a random person and measuring their height yields another random variable.

Mathematically, a random variable is defined as a [[measurable function]] from a [[probability space]] to some [[measurable space]]. This measurable space is the space of possible values of the variable, and it is usually taken to be the [[real number|real numbers]] with the [[Borel algebra|Borel σ-algebra]], and we will always assume this in this encyclopedia, unless otherwise specified. This is a technical device used to guarantee the existence of random variables, and sometimes to construct them. In practice, one disposes of the space Ω altogether and just puts a measure on '''R''' that assigns measure 1 to the whole real line.

If a random variable ''X'':Ω->'''R''' defined on the probability space (Ω, ''P'') is given, we can ask questions like "How likely is it that the value of ''X'' is bigger than 2?". This is the same as the probability of the event {''s'' in Ω : ''X''(''s'') > 2} which is often written as ''P''(''X'' > 2) for short.

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:''F_X''(x) = P(X≤x)

and sometimes also using a [[probability density function]]. In [[measure theory|measure-theoretic]] terms, we use the random variable ''X'' to "push-forward" the measure ''P'' on Ω to a measure d''F'' on '''R'''.

The underlying probability space Ω is a technical device used to guarantee the existence of random variables, and sometimes to construct them. In practice, one often disposes of the space Ω altogether and just puts a measure on '''R''' that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables.

A random variable can often be characterised by a small number of quantities, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. For instance, what is the average of the results you get when you roll a die represtedly, or measure human heights? This is captured by the mathematical concept of [[expected value]] of a random variable, denoted E[''X'']. Once the "average value" is known, one could then ask "how far are the values of ''X'' typically away from the average", a question that is answered by the [[variance]] and [[standard deviation]] of a random variable.

Mathematically, this is calledknown as the (generalised) [[problem of moments]]: tofor a given class of random variables ''X'', find a collection {''f_i''} of functions of ''X'' such that theirthe expectation values E[''f_i''(''X'')] fully characterize the distribution of the random variable ''X''.

See also: [[discrete random variable]], [[continuous random variable]], [[probability distribution]]