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1.2 Expectation Values

An important definition which we will make use of later is the expectation value of a random variable or a random variable function. If x is a random variable distributed as P(x), then

Equation 5 (5)

is the expected value of x. The integration in (5) is over all admissible x. This, of course, is just the standard notion of an average value. For a discrete variable, (5) becomes a sum

Equation 6 (6)

Similarly, if f(x) is a function of x, then

Equation 7 (7)

is the expected value of f(x).

To simplify matters in the remainder of this section, we will present results assuming a continuous variable. Unless specified otherwise, the discrete case is found by replacing integrals with a summation.