Expected Value

In probability theory, the expected value (a.k.a. (mathematical) expectation, mean, average, or first moment) is a generalization of the weighted average. 
The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes, E[X] = \sum_{i \in X} x_i p_i. When the set of possible outcomes is not countable, but a continuum, the expectation is defined by integration instead of summation E[X] = \int_{-\infty}^\infty x f(x) dx. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration.
== Notation ==
The expected value of a random variable X is often denoted byE(X), E[X], EX, \mathbb{E}[X], or \mu_X, \bar{X} where the latter are more common in physics to denote the mean. 
== Definition ==
Put in the form of Expectation under different possibilities of the set X.