Random variables

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable (i.e., not necessarily fixed) quantity whose possible values depend, in some clearly-defined way, on a set of random . When the random variable is discrete, it can take on a value from a discrete set of possible different values, each with an associated probability. Like a traditional mathematical , its value is thus unknown a priori (before the outcome of the events is known).

In a more abstract sense, a random variable is defined as a that maps outcomes (that is, points in a probability space) to mathematically convenient outcome labels, usually real numbers. In this sense, it is a procedure for assigning a number to an outcome, and, contrary to its name, this procedure itself is neither random nor variable. The function which characterizes a random variable must also be measurable, which rules out certain pathological cases such as those in which the random variable's quantity is infinitely sensitive to any small change in the outcome.

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, due to imprecise measurements or quantum uncertainty). 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. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself but is instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.

A random variable has a probability distribution, which specifies the probability that its value falls in any given interval. 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. Two random variables with the same probability distribution can still differ in terms of their associations with, or from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.

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