Random variable


A random variable is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which
Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup.
In the formal mathematical language of measure theory, a random variable is defined as a measurable function from a probability measure space to a measurable space. This allows consideration of the pushforward measure, which is called the distribution of the random variable; the distribution is thus a probability measure on the set of all possible values of the random variable. It is possible for two random variables to have identical distributions but to differ in significant ways; for instance, they may be independent.
It is common to consider the special cases of discrete random variables and absolutely continuous random variables, corresponding to whether a random variable is valued in a countable subset or in an interval of real numbers. There are other important possibilities, especially in the theory of stochastic processes, wherein it is natural to consider random sequences or random functions. Sometimes a random variable is taken to be automatically valued in the real numbers, with more general random quantities instead being called random elements.
According to George Mackey, Pafnuty Chebyshev was the first person "to think systematically in terms of random variables".

Definition

A random variable is a measurable function from a sample space as a set of possible outcomes to a measurable space. For the measurability of to be meaningful, the sample space needs to belong to a probability triple . A random variable is often denoted by capital Roman letters such as.
The probability that takes on a value in a measurable set is written as

Standard case

In many cases, is real-valued, i.e.. In some contexts, the term random element is used to denote a random variable not of this form.
When the image of is finite or countably infinite, the random variable is called a discrete random variable and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability to each value in the image of. If the image is uncountably infinite then 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.
Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value.

Extensions

The term "random variable" in statistics is traditionally limited to the real-valued case. In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.
However, the definition above is valid for any measurable space of values. Thus one can consider random elements of other sets, such as random Boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions. One may then specifically refer to a random variable of type , or an -valued random variable.
This more general concept of a random element is particularly useful in disciplines such as graph theory, machine learning, natural language processing, and other fields in discrete mathematics and computer science, where one is often interested in modeling the random variation of non-numerical data structures. In some cases, it is nonetheless convenient to represent each element of, using one or more real numbers. In this case, a random element may optionally be represented as a vector of real-valued random variables. For example:
  • A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are,, and the position of the 1 indicates the word.
  • A random sentence of given length may be represented as a vector of random words.
  • A random graph on given vertices may be represented as a matrix of random variables, whose values specify the adjacency matrix of the random graph.
  • A random function may be represented as a collection of random variables, giving the function's values at the various points in the function's domain. The are ordinary real-valued random variables provided that the function is real-valued. For example, a stochastic process is a random function of time, a random vector is a random function of some index set such as, and a random field is a random function on any set.

    Distribution functions

If a random variable defined on the probability space is given, we can ask questions like "How likely is it that the value of is equal to 2?". This is the same as the probability of the event which is often written as or for short.
Recording all these probabilities of outputs of a random variable yields the probability distribution of. The probability distribution "forgets" about the particular probability space used to define and only records the probabilities of various output values of. Such a probability distribution, if is real-valued, can always be captured by its cumulative distribution function
and sometimes also using a probability density function,. In measure-theoretic terms, we use the random variable to "push-forward" the measure on to a measure on. The measure is called the " distribution of " or the "law of ".
The density, the Radon–Nikodym derivative of with respect to some reference measure on .
The underlying probability space 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 altogether and just puts a measure on that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on quantile functions for fuller development.

Examples

Discrete random variable

Consider an experiment where a person is chosen at random. An example of a random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to their height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the 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 – 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.
In examples such as these, the sample space 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.
If are countable sets of real numbers, and, then is a discrete distribution function. Here for, for. Taking for instance an enumeration of all rational numbers as, one gets a discrete function that is not necessarily a step function.

Coin toss

The possible outcomes for one coin toss can be described by the sample space. We can introduce a real-valued random variable that models a $1 payoff for a successful bet on heads as follows:
If the coin is a fair coin, has a probability mass function given by:

Dice roll

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 as the sample space. The total number rolled is then a random variable X given by the function that maps the pair to the sum:
and has a probability mass function fX given by: