Definition
A random variable is defined on a set of possible outcomes (the sample space Ω) and a probability distribution that associates each outcome with a probability. A random variable represents a measurable aspect or property of the outcomes, and hence associates each outcome with a number. In an experiment a person may be chosen at random, and one random variable may be its age, and another its number of children. Formally a random variable is considered to be a function on the possible outcomes. Random variables are typically classified as either discrete or continuous. Discrete variables can take on either a finite or at most a countably infinite set of discrete values. Their probability distribution is given by a probability mass function which directly maps a value of the random variable to a probability. Continuous variables, however, take on values that vary continuously within one or more (possibly infinite) intervals. As a result there are an uncountably infinite number of individual outcomes, and each has a probability 0. As a result, the probability distribution for many continuous random variables is defined using a probability density function, which indicates the "density" of probability in a small neighborhood around a given value. More technically, the probability that an outcome is in a particular range is derived from the integration of the probability density function in that range. Both concepts can be united using a cumulative distribution function (CDF), which describes the probability that an outcome will be less than or equal to a specified value. This function is necessarily monotonically non-decreasing, with a minimum value of 0 at negative infinity and a maximum value of 1 at positive infinity. The CDF of a discrete distribution will consist mostly of flat areas along with sudden jumps at each outcome defined in the sample space, while the CDF of a continuous distribution will typically rise gradually and continuously. Distributions that are partly discrete and partly continuous can also be described this way.
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