Discrete random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable quantity whose value depends, in a random manner, on a set of possible outcomes. It is common that these outcomes depend on some physical variables that are not well understood. For example, when you toss a coin, the final outcome of heads or tails depends on the uncertain physics. Which outcome will be observed is not certain. Of course the coin could get caught in a crack in the floor, but such a possibility is excluded from consideration. Part of the definition of a random variable is the set of possible outcomes. In the case of the coin, there are only two possible outcomes, namely heads or tails. Thus either heads or tails must land with non-zero probability.

A random variable is defined as a function that maps outcomes to numerical quantities (labels), typically real numbers. In this sense, it is a procedure for assigning a numerical quantity to each physical outcome, and, contrary to its name, this procedure itself is neither random nor variable. What is random is the unstable physics that describes how the coin lands, and the uncertainty of which outcome will actually be observed.

As a function, a random variable is required to be measurable, which rules out certain pathological cases where the quantity which the random variable returns is infinitely sensitive to small changes 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.

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