 In probability and statistics, the random variable, random quantities, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon. As a function, the 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. It is common that these outcomes depend on some physical variables that are not well understood. For example, when talking a fair coin, the final outcome of heads or tails depends on the uncertain physics. 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. The domain 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. Since one of these outcomes must occur, either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability. The random variable is defined as a function that maps the outcomes of unpredictable processes to numerical quantities labels typically real numbers. In this sense, it is a procedure for assigning a numerical quantity to each physical outcome. Contrary to its name, this procedure itself is neither random nor variable. Rather, the underlying process providing the input to this procedure yields random possibly non-numerical output that the procedure maps to a real numbered value. The random variables 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 brolling 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. The 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 the 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 independents 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 variants. The formal mathematical treatment of random variables is a topic in probability theory. In that context, the random variable is understood as a function defined on a sample space whose outputs are numerical values.