The expected value is the mean of all values of a random variable weighted by the probability of the occurrence of the values. The expected value (or expectation, or mean) of random variable (RV) X is denoted as E[X] (or sometimes μ).
The RV, X, of a random experiment, which is defined on a probability space (Ω, Σ, P) on an underlying sample space Ω, takes value in event set Σ ⊆ R with certain probability measure P.
If X is a continuous random variable (i.e., Σ is an interval), the expected value of X is defined as
If X also has a probability density function (pdf) f(x) of certain probability distribution, the above expected value of X can be formulated as
The expected value exists if the ...