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.
Read more about this topic: Random Variable
Famous quotes containing the word definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)