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In probability theory and statistics, a random variable is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a sample space whose outcomes are typically real numbers.

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, because of imprecise measurements). 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.

Random variables as functions

As a function, a random variable is required to be measurable, which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions. Which outcome will be observed is not certain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.

Discrete and continuous random variables

A random variable has a probability distribution, which specifies the probability of its values. 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 the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both types.

See also