Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a <a href="/facts/Mathematics/pxTouaz4"> mathematical</a> formalization of a quantity or object which depends on <a href="/facts/Randomness/QJQBoEX5">random</a> events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> in which 

<ul><li>the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> is the set of possible <a href="/facts/Outcome_(probability)/1hdJcoQR">outcomes</a> in a <a href="/facts/Sample_space/WVldkN4F">sample space</a> (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</li>
<li>the <a href="/facts/Range_of_a_function/66CeTV6u">range</a> is a <a href="/facts/Measurable_space/AJ98lhT4">measurable space</a> (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 <a href="/facts/Real_number/R02gw5Pb">real numbers</a>.</li></ul>

Informally, randomness typically represents some fundamental element of chance, such as in the roll of a <a href="/facts/Dice/qBnylzwJ">die</a>; it may also represent uncertainty, such as <a href="/facts/Measurement_error/q7pa7bRt">measurement error</a>. However, the <a href="/facts/Interpretation_of_probability/qmuEgkIU">interpretation of probability</a> 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 <a href="/facts/Axiom/mLjP14Mc">axiomatic</a> setup.
In the formal mathematical language of <a href="/facts/Measure_theory/yCq7zlI4">measure theory</a>, a random variable is defined as a <a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> from a <a href="/facts/Probability_measure_space/8LJINLNc">probability measure space</a> (called the sample space) to a <a href="/facts/Measurable_space/AJ98lhT4">measurable space</a>. This allows consideration of the <a href="/facts/Pushforward_measure/aFljwqr6">pushforward measure</a>, which is called the distribution of the random variable; the distribution is thus a <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a> 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 <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a>.
It is common to consider the special cases of <a href="/facts/Discrete_random_variable/TwTBXnLT">discrete random variables</a> and <a href="/facts/Probability_distribution/EpsKKVRu">absolutely continuous random variables</a>, corresponding to whether a random variable is valued in a countable subset or in an interval of <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. There are other important possibilities, especially in the theory of <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic processes</a>, wherein it is natural to consider <a href="/facts/Random_sequence/z2uHfvK4">random sequences</a> or <a href="/facts/Random_function/Ng1n1dnB">random functions</a>. Sometimes a random variable is taken to be automatically valued in the real numbers, with more general random quantities instead being called <a href="/facts/Random_element/y0H37fQx">random elements</a>.
According to <a href="/facts/George_Mackey/uoeY23Qv">George Mackey</a>, <a href="/facts/Pafnuty_Chebyshev/LzRqLqRm">Pafnuty Chebyshev</a> was the first person "to think systematically in terms of random variables".

Random variable open-in-new

Random variable