Random variable

In probability and statistics, a random variable, aleatory variable or stochastic variable is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense).[1]:391 A random variable can take on a set of possible different values (similarly to other mathematical variables), each with an associated probability, in contrast to other mathematical variables.

A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, due to imprecise measurements or quantum uncertainty). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself but is instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.

The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution. Random variables can be discrete, that is, taking any of a specified finite or countable list of values, endowed with a probability mass function, characteristic of a probability distribution; or continuous, taking any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of a probability distribution; or a mixture of both types. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.

The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a function defined on a sample space whose outputs are numerical values.[2]

Definition

A random variable X\colon \Omega \to E is a measurable function from the set of possible outcomes \Omega to some set E. Usually, E = \mathbb{R}, otherwise the term random element is used instead (see Extensions). The technical axiomatic definition requires both \Omega and E to be measurable spaces (see Measure-theoretic definition).

As a real-valued function, X often describes some numerical quantity of a given event. E.g. the number of heads after a certain number of coin flips; the heights of different people.

When the image (or range) of X is finite or countably infinite, the random variable is called a discrete random variable[1]:399 and its distribution can be described by a probability mass function which assigns a probability to each value in the image of X. If the image is uncountably infinite then X is called a continuous random variable. In the special case that it is absolutely continuous, its distribution can be described by a probability density function, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous,[3] for example a mixture distribution. Such random variables cannot be described by a probability density or a probability mass function.

All random variables can be described by their cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value.

Extensions

The basic concept of "random variable" in statistics is real-valued, and therefore expected values, variances and other measures can be computed. However, one can consider arbitrary types such as boolean values, categorical variables, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, functions, and processes. The term random element is used to encompass all such related concepts.

Another extension is the stochastic process, a set of indexed random variables (typically indexed by time or space).

These more general concepts are particularly useful in fields such as computer science and natural language processing where many of the basic elements of analysis are non-numerical. Such general random elements can sometimes be treated as sets of real-valued random variables — often more specifically as random vectors. For example:

Reduction to numerical values is not essential for dealing with random elements: a randomly selected individual remains an individual, not a number.

Examples

Discrete random variable

In an experiment a person may be chosen at random, and one random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to the person's height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any non-pathological subset of possible values, such as probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.

Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum PMF(0) + PMF(2) + PMF(4) + ...

In examples such as these, the sample space (the set of all possible persons) is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.

Coin Toss

The possible outcomes for one coin toss can be described by the sample space \Omega = \{\text{heads}, \text{tails}\}. We can introduce a real-valued random variable Y that models a $1 payoff for a successful bet on heads as follows:

Y(\omega) = \begin{cases} 1, & \text{if} \ \ \omega = \text{heads} ,\\ \\ 0, & \text{if} \ \ \omega = \text{tails} . \end{cases}

If the coin is a fair coin, Y has a probability mass function f_Y given by:

f_Y(y) = \begin{cases} \tfrac 12,& \text{if }y=1,\\ \\ \tfrac 12,& \text{if }y=0,\\ \end{cases}

Die Roll

If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sum S of the numbers on the two dice, then S is a discrete random variable whose distribution is described by the probability mass function plotted as the height of picture columns here.

A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers n1 and n2 from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable X given by the function that maps the pair to the sum:

X((n_1, n_2)) = n_1 + n_2

and (if the dice are fair) has a probability mass function ƒX given by:

f_X(S) = \tfrac{\min(S-1, 13-S)}{36}, \text{for } S \in \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}

Continuous Random Variable

An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, X = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any range of values. For example, the probability of choosing a number in [0, 180] is 1⁄2. Instead of speaking of a probability mass function, we say that the probability density of X is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.

Mixed type

An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, X = −1; otherwise X = the value of the spinner as in the preceding example. There is a probability of 1⁄2 that this random variable will have the value −1. Other ranges of values would have half the probability of the last example.

Measure-theoretic definition

The most formal, axiomatic definition of a random variable involves measure theory. Continuous random variables are defined in terms of sets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the Banach–Tarski paradox) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a sigma-algebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the Borel σ-algebra, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals.[2]

The measure-theoretic definition is as follows.

Let (\Omega, \mathcal{F}, P) be a probability space and (E, \mathcal{E}) a measurable space. Then an (E, \mathcal{E})-valued random variable is a function X\colon \Omega \to E which is (\mathcal{F}, \mathcal{E})-measurable. The latter means that, for every subset B\in\mathcal{E}, its preimage X^{-1}(B)\in \mathcal{F} where X^{-1}(B) = \{\omega : X(\omega)\in B\}.[4] This definition enables us to measure any subset B\in \mathcal{E} in the target space by looking at its preimage, which by assumption is measurable.

When E is a topological space, then the most common choice for the σ-algebra \mathcal{E} is the Borel σ-algebra \mathcal{B}(E), which is the σ-algebra generated by the collection of all open sets in E. In such case the (E, \mathcal{E})-valued random variable is called the E-valued random variable. Moreover, when space E is the real line \mathbb{R}, then such real-valued random variable is called simply the random variable.

Real-valued random variables

In this case the observation space is the real numbers. Recall, (\Omega, \mathcal{F}, P) is the probability space. For real observation space, the function X\colon \Omega \rightarrow \mathbb{R} is a real-valued random variable if

\{ \omega : X(\omega) \le r \} \in \mathcal{F} \qquad \forall r \in \mathbb{R}.

This definition is a special case of the above because the set \{(-\infty, r]: r \in \R\} generates the Borel σ-algebra on the real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that \{ \omega : X(\omega) \le r \} = X^{-1}((-\infty, r]).

Distribution functions of random variables

If a random variable X\colon \Omega \to \mathbb{R} defined on the probability space (\Omega, \mathcal{F}, P) is given, we can ask questions like "How likely is it that the value of X is equal to 2?". This is the same as the probability of the event \{ \omega : X(\omega) = 2 \}\,\! which is often written as P(X = 2)\,\! or 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 \le x)

and sometimes also using a probability density function, p_X. In measure-theoretic terms, we use the random variable X to "push-forward" the measure P on \Omega to a measure p_X on \mathbb{R}. The underlying probability space \Omega is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on a joint distribution of two or more random variables on the same probability space. In practice, one often disposes of the space \Omega altogether and just puts a measure on \mathbb{R} that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables.

Moments

The probability distribution of a 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], and also called the first moment. In general, E[f(X)] is not equal to 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. E[X] can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of X.

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 characterise the distribution of the random variable X.

Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function f(X)=X of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable X that can take on the nominal values "red", "blue" or "green", the real-valued function [X = \text{green}] can be constructed; this uses the Iverson bracket, and has the value 1 if X has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined.

Functions of random variables

A new random variable Y can be defined by applying a real Borel measurable function g\colon \mathbb{R} \rightarrow \mathbb{R} to the outcomes of a real-valued random variable X. The cumulative distribution function of Y\,\! is

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

If function g is invertible, i.e. g−1 exists, and is either increasing or decreasing, then the previous relation can be extended to obtain

F_Y(y) = \operatorname{P}(g(X) \le y) = \begin{cases} \operatorname{P}(X \le g^{-1}(y)) = F_X(g^{-1}(y)), & \text{if } g^{-1} \text{ increasing} ,\\ \\ \operatorname{P}(X \ge g^{-1}(y)) = 1 - F_X(g^{-1}(y)), & \text{if } g^{-1} \text{ decreasing} . \end{cases}

and, again with the same hypotheses of invertibility of g, assuming also differentiability, we can find the relation between the probability density functions by differentiating both sides with respect to y, in order to obtain

f_Y(y) = f_X(g^{-1}(y)) \left| \frac{d g^{-1}(y)}{d y} \right|.

If there is no invertibility of g but each y admits at most a countable number of roots (i.e. a finite, or countably infinite, number of xi such that y = g(xi)) then the previous relation between the probability density functions can be generalized with

f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right|

where xi = gi−1(y). The formulas for densities do not demand g to be increasing.

In the measure-theoretic, axiomatic approach to probability, if we have a random variable X\! on \Omega \,\! and a Borel measurable function g\colon \mathbb{R} \rightarrow \mathbb{R}, then Y = g(X)\,\! will also be a random variable on \Omega\,\! , since the composition of measurable functions is also measurable. (However, this is not true if g is Lebesgue measurable.) The same procedure that allowed one to go from a probability space (\Omega, P)\,\! to (\mathbb{R}, dF_{X}) can be used to obtain the distribution of Y\,\! .

Example 1

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

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

\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.

Example 2

Suppose \scriptstyle X is a random variable with a cumulative distribution

F_{X}(x) = P(X \leq x) = \frac{1}{(1 + e^{-x})^{\theta}}

where \scriptstyle \theta > 0 is a fixed parameter. Consider the random variable \scriptstyle Y = \mathrm{log}(1 + e^{-X}). Then,

F_{Y}(y) = P(Y \leq y) = P(\mathrm{log}(1 + e^{-X}) \leq y) = P(X > -\mathrm{log}(e^{y} - 1)).\,

The last expression can be calculated in terms of the cumulative distribution of X, so

F_{Y}(y) = 1 - F_{X}(-\mathrm{log}(e^{y} - 1)) \,
= 1 - \frac{1}{(1 + e^{\mathrm{log}(e^{y} - 1)})^{\theta}}
= 1 - \frac{1}{(1 + e^{y} - 1)^{\theta}}
= 1 - e^{-y \theta}.\,

Example 3

Suppose \scriptstyle X is a random variable with a standard normal distribution, whose density is

f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}.

Consider the random variable \scriptstyle Y = X^2. We can find the density using the above formula for a change of variables:

f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right|.

In this case the change is not monotonic, because every value of \scriptstyle Y has two corresponding values of \scriptstyle X (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.

f_Y(y) = 2f_X(g^{-1}(y)) \left| \frac{d g^{-1}(y)}{d y} \right|.

The inverse transformation is

x = g^{-1}(y) = \sqrt{y}

and its derivative is

\frac{d g^{-1}(y)}{d y} = \frac{1}{2\sqrt{y}} .

Then:

f_Y(y) = 2\frac{1}{\sqrt{2\pi}}e^{-y/2} \frac{1}{2\sqrt{y}} = \frac{1}{\sqrt{2\pi y}}e^{-y/2}.

This is a chi-squared distribution with one degree of freedom.

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, or equal in distribution.

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

Equality in distribution

If the sample space is a subset of the real line, random variables X and Y are equal in distribution (denoted X \stackrel{d}{=} Y) 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 i.i.d. random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform.

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)=\mathrm{ess } \sup_\omega|X(\omega)-Y(\omega)|,

where "ess sup" represents the essential supremum in the sense of measure theory.

Equality

Finally, the two random variables X and Y are equal if they are equal as functions on their measurable space:

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

Convergence

A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; 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

References

  1. ↑ 1.0 1.1 Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003). The Practice of Statistics (2nd ed.). New York: Freeman. ISBN 978-0-7167-4773-4.
  2. ↑ 2.0 2.1 Steigerwald, Douglas G. "Economics 245A – Introduction to Measure Theory" (PDF). University of California, Santa Barbara. Retrieved April 26, 2013.
  3. ↑ L. Castañeda, V. Arunachalam, and S. Dharmaraja (2012). Introduction to Probability and Stochastic Processes with Applications. Wiley. p. 67.
  4. ↑ Fristedt & Gray (1996, page 11)

Literature

External links