A random variable (also called random quantity, aleatory variable, or stochastic 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
the domain is the set of possible outcomes in a sample space (e.g. the set { H , T } {\displaystyle \{H,T\}} (which are the possible upper sides of a flipped coin heads H {\displaystyle H} or tails T {\displaystyle T} as the result from tossing a coin); and the range is a measurable space (e.g. corresponding to the domain above, the range might be the set { − 1 , 1 } {\displaystyle \{-1,1\}} if say heads H {\displaystyle H} mapped to −1 and T {\displaystyle T} mapped to 1). Typically, the range of a random variable is a subset of the real numbers.
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 (called the sample 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. A random variate is a particular outcome or realization of a random variable. According to George Mackey, Pafnuty Chebyshev was the first person "to think systematically in terms of random variables".
Definition A random variable X {\displaystyle X} is a measurable function X : Ω → E {\displaystyle X\colon \Omega \to E} from a sample space Ω {\displaystyle \Omega } as a set of possible outcomes to a measurable space E {\displaystyle E} . For the measurability of X {\displaystyle X} to be meaningful, the sample space Ω {\displaystyle \Omega } needs to belong to a probability triple ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )} (see the measure-theoretic definition). A random variable is often denoted by capital Roman letters such as X , Y , Z , T {\displaystyle X,Y,Z,T} . The probability that X {\displaystyle X} takes on a value in a measurable set S ⊆ E {\displaystyle S\subseteq E} is written as
P ( X ∈ S ) = P ( { ω ∈ Ω ∣ X ( ω ) ∈ S } ) {\displaystyle \operatorname {P} (X\in S)=\operatorname {P} (\{\omega \in \Omega \mid X(\omega )\in S\})} .
Standard case In many cases, X {\displaystyle X} is real-valued, i.e. E = R {\displaystyle E=\mathbb {R} } . In some contexts, the term random element (see extensions) is used to denote a random variable not of this form. When the image (or range) of X {\displaystyle X} 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 X {\displaystyle X} . If the image is uncountably infinite (usually an interval) then X {\displaystyle 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. 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.
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