Expectation operator

Averaging across a probability distribution is so common in Statistics that a special notation has been developed to denote it. This is the expectation operator. The formal definition of the expectation of a function g applied to a random variable X is given by (for a continuous random variable)

• E_X[g(x)] = INTEGRAL g(x) dF(x) where dF indicates the probability element at x and the region of integration includes all values of the variable with non-zero probability density.

In the case of discrete random variables, we have

• E_X[g(x)] = SUM g(x) p(x) where p(x) is the probability distribution of the random variable X.

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