Random variable

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A random variable is a term used in mathematics and statistics. It can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. For example, a random variable can be used to describe the process of rolling a fair die and the possible outcomes { 1, 2, 3, 4, 5, 6 }. Another random variable might describe the possible outcomes of picking a random person and measuring his or her height.

Unlike the common practice with other mathematical variables, a random variable cannot be assigned a value; a random variable does not describe the actual outcome of a particular experiment, but rather describes the possible, as-yet-undetermined outcomes in terms of real numbers.

Although such simple examples as rolling a die and measuring heights allow easy visualisation of the practical use of random variables, their mathematical construction allows mathematicians the convenience of dealing with much measure-theoretic probability theory in the more familiar domain of real-valued functions. Conversely, the concept also places experiments involving real-valued outcomes firmly within the measure-theoretic framework.

Definitions[]

Random variables[]

Some consider the expression random variable a misnomer, as a random variable is not a variable but rather a function that maps events to numbers. Let A be a σ-algebra and Ω the space of events relevant to the experiment being performed. In the die-rolling example, the space of events is just the possible outcomes of a roll, i.e. Ω = { 1, 2, 3, 4, 5, 6 }, and A would be the power set of Ω. In this case, an appropriate random variable might be the identity function X(ω) = ω, such that if the outcome is a '1', then the random variable is also equal to 1. An equally simple but less trivial example is one in which we might toss a coin: a suitable space of possible events is Ω = { H, T } (for heads and tails), and A equal again to the power set of Ω. One among the many possible random variables defined on this space is

X(\omega) = \begin{cases}0,& \omega = \texttt{H},\\1,& \omega = \texttt{T}.\end{cases}

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 numbers with the Borel σ-algebra. This is assumed in the following, except where specified.

Let (Ω, A, P) be a probability space. Formally, a function X: Ω → R is a (real-valued) random variable if for every subset A_r = { ω : X(ω) ≤ r } where r ∈ R, we also have A_r ∈ A. The importance of this technical definition is that it allows us to construct the distribution function of the random variable.

Distribution functions[]

If a random variable $X:\Omega \to \mathbb{R}$ defined on the probability space $(\Omega , P)$ 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 $\{ s \in\Omega : X(s) > 2 \}$ which is often written as $P(X > 2)$ for short.

Recording all these probabilities of output ranges of a real-valued random variable X yields the probability distribution of X. The probability distribution "forgets" about the particular probability space used to define X and only records the probabilities of various values of X. Such a probability distribution can always be captured by its cumulative distribution function

F_X(x) = \operatorname{P}(X\leq x)

and sometimes also using a probability density function. In measure-theoretic terms, we use the random variable X to "push-forward" the measure P on Ω to a measure dF 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.

Functions of random variables[]

If we have a random variable X on Ω and a measurable function f: R → R, then Y = f(X) will also be a random variable on Ω, since the composition of measurable functions is also measurable. The same procedure that allowed one to go from a probability space (Ω, P) to (R, dF_X) can be used to obtain the distribution of Y. The cumulative distribution function of Y is

F_Y(y) = \operatorname{P}(f(X) \le y).

Example[]

Let X be a real-valued, continuous random variable and let Y = X². Then,

F_Y(y) = \operatorname{P}(X^2 \le y).

If y < 0, then P(X² ≤ y) = 0, so

F_Y(y) = 0\qquad\hbox{if}\quad y < 0.

If y ≥ 0, then

{\displaystyle \operatorname{P}(X^2 \le y) = \operatorname{P}(|X| \le \sqrt{y}) = \operatorname{P}(-\sqrt{y} \le X \le \sqrt{y}),}

so

F_Y(y) = F_X(\sqrt{y}) - F_X(-\sqrt{y})\qquad\hbox{if}\quad y \ge 0.

Moments[]

The probability distribution of random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted E[X]. Note that in general, E[f(X)] is not the same as f(E[X]). Once the "average value" is known, one could then ask how far from this average value the values of X typically are, a question that is answered by the variance and standard deviation of a random variable.

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

Equivalence of random variables[]

There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, equal in mean, or equal in distribution.

In increasing order of strength, the precise definition of these notions of equivalence is given below.

Equality in distribution[]

Two random variables X and Y are equal in distribution if they have the same distribution functions:

\operatorname{P}(X \le x) = \operatorname{P}(Y \le x)\quad\hbox{for all}\quad x.

Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of iidrv's.

To be equal in distribution, random variables need not be defined on the same probability space. The notion of equivalence in distribution is associated to the following notion of distance between probability distributions,

d(X,Y)=\sup_x|\operatorname{P}(X \le x) - \operatorname{P}(Y \le x)|,

which is the basis of the Kolmogorov-Smirnov test.

Equality in mean[]

Two random variables X and Y are equal in p-th mean if the pth moment of |X − Y| is zero, that is,

\operatorname{E}(|X-Y|^p) = 0.

Equality in pth mean implies equality in qth mean for all q<p. As in the previous case, there is a related distance between the random variables, namely

d_p(X, Y) = \operatorname{E}(|X-Y|^p).

Almost sure equality[]

Two random variables X and Y are equal almost surely if, and only if, the probability that they are different is zero:

\operatorname{P}(X \neq Y) = 0.

For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:

d_\infty(X,Y)=\sup_\omega|X(\omega)-Y(\omega)|,

where 'sup' in this case represents the essential supremum in the sense of measure theory.

Equality[]

Finally, two random variables X and Y are equal if they are equal as functions on their probability space, that is,

X(\omega)=Y(\omega)\qquad\hbox{for all}\quad\omega

Convergence[]

Much of mathematical statistics consists in proving convergence results for certain sequences of random variables; see for instance the law of large numbers and the central limit theorem.

There are various senses in which a sequence (X_n) of random variables can converge to a random variable X. These are explained in the article on convergence of random variables.

Literature[]

Papoulis, Athanasios 1965 Probability, Random Variables, and Stochastic Processes. McGraw-Hill Kogakusha, Tokyo, 9th editon, ISBN 0071199810.