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Random variable

From Academic Kids

A random variable 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 their 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.

Contents

Definitions

Random variables

As noted above, a random variable is essentially a function mapping events to numbers. Let A be a σ-algebra and Ω be 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 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

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

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 Ar = { ω : X(ω) ≤ r } where rR, we also have ArA.

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 <math>X: \Omega \to \mathbb{R}<math> defined on the probability space <math>(\Omega , P)<math> is given, we can ask questions like "How likely is it that the value of <math>X<math> is bigger than 2?". This is the same as the probability of the event <math>\{ s \in\Omega : X(s) > 2 \} <math> which is often written as <math>P(X > 2)<math> 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

<math>F_X(x) = \operatorname{P}(X < x)<math>

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: RR, 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, dFX) can be used to obtain the distribution of Y. The cumulative distribution function of Y is

<math>F_Y(y) = \operatorname{P}(f(X) < y).<math>

Example

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

<math>F_Y(y) = \operatorname{P}(X^2 < y).<math>

If y < 0, then P(X2y) = 0, so

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

If y ≥ 0, then

<math>\operatorname{P}(X^2 < y) = \operatorname{P}(|X| < \sqrt{y})
= \operatorname{P}(-\sqrt{y} <  X < \sqrt{y}),<math>

so

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

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 {fi} of functions such that the expectation values E[fi(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

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

To be equal in distribution, random variables need not be defined on the same probability space, but without loss of generality they can be made into independent random variables on the same probability space. The notion of equivalence in distribution is associated to the following notion of distance between probability distributions,

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

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 |XY| is zero, that is,

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

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

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

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:

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

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

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

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,

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

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 (Xn) of random variables can converge to a random variable X. These are explained in the article on convergence of random variables.


See also:

discrete random variable, continuous random variable, probability distribution, randomness, random vector, random function, generating function. Algorithmic information theory


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