In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or the first moment) of a random variable is the weighted average of all possible values that this random variable can take on. The weights used in computing this average correspond to the probabilities in case of a discrete random variable, or densities in case of a continuous random variable. From a rigorous theoretical standpoint, the expected value is the integral of the random variable with respect to its probability measure.
The expected value may be intuitively understood by the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity. More informally, it can be interpreted as the long-run average of the results of many independent repetitions of an experiment (e.g. a dice roll). The value may not be expected in the ordinary sense—the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), just like the sample mean.
The expected value does not exist for some distributions with large "tails", such as the Cauchy distribution.
Read more about Expected Value: History, Uses and Applications, Expectation of Matrices
Famous quotes containing the word expected:
“Perhaps of all our untamed quadrupeds, the fox has obtained the widest and most familiar reputation.... His recent tracks still give variety to a winter’s walk. I tread in the steps of the fox that has gone before me by some hours, or which perhaps I have started, with such a tip-toe of expectation as if I were on the trail of the Spirit itself which resides in the wood, and expected soon to catch it in its lair.”
—Henry David Thoreau (1817–1862)