Random variable

Lore & Background

According to George Mackey, Pafnuty Chebyshev was the first person 'to think systematically in terms of random variables.' The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function. The domain is the set of possible outcomes in a sample space, and the range is a measurable space, typically a subset of the real numbers. Informally, randomness represents chance or uncertainty, but the mathematical analysis is independent of interpretational difficulties and based on rigorous axiomatic setups.

Reader's Guide

In the formal language of measure theory, a random variable is defined as a measurable function from a probability measure space (the sample space) to a measurable space. This allows consideration of the pushforward measure, called the distribution of the random variable. It is common to consider discrete random variables (valued in a countable subset) and absolutely continuous random variables (valued in an interval of real numbers). Any random variable can be described by its cumulative distribution function. The term 'random variable' in statistics is traditionally limited to real-valued cases, but the definition is valid for any measurable space, allowing random elements such as Boolean values, vectors, functions, and more. This broader concept is useful in fields like graph theory, machine learning, and computer science.

Did You Know?

• A random variable is a measurable function from a sample space to a measurable space.
• Two random variables can have identical distributions but differ in significant ways, such as being independent.
• Pafnuty Chebyshev was the first person to think systematically in terms of random variables, according to George Mackey.
• A random variate is a particular outcome or realization of a random variable.