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# Expectation Value

The expectation value of a function in a variable is denoted or . For a single discrete variable, it is defined by

 (1)

where is the probability density function.

For a single continuous variable it is defined by,

 (2)

The expectation value satisfies

 (3) (4) (5)

For multiple discrete variables

 (6)

For multiple continuous variables

 (7)

The (multiple) expectation value satisfies

 (8) (9) (10)

where is the mean for the variable .

Central Moment, Estimator, Maximum Likelihood, Mean, Moment, Raw Moment, Wald's Equation

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## References

Papoulis, A. "Expected Value; Dispersion; Moments." §5-4 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 139-152, 1984.

## Referenced on Wolfram|Alpha

Expectation Value

## Cite this as:

Weisstein, Eric W. "Expectation Value." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExpectationValue.html