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Expectation Value


The expectation value of a function f(x) in a variable x is denoted <f(x)> or E{f(x)}. For a single discrete variable, it is defined by

 <f(x)>=sum_(x)f(x)P(x),
(1)

where P(x) is the probability density function.

For a single continuous variable it is defined by,

 <f(x)>=intf(x)P(x)dx.
(2)

The expectation value satisfies

<ax+by>=a<x>+b<y>
(3)
<a>=a
(4)
<sumx>=sum<x>.
(5)

For multiple discrete variables

 <f(x_1,...,x_n)>=sum_(x_1,...,x_n)f(x_1,...,x_n)P(x_1,...,x_n).
(6)

For multiple continuous variables

 <f(x_1,...,x_n)>=intf(x_1,...,x_n)P(x_1,...,x_n)dx_1...dx_n.
(7)

The (multiple) expectation value satisfies

<(x-mu_x)(y-mu_y)>=<xy-mu_xy-mu_yx+mu_xmu_y>
(8)
=<xy>-mu_xmu_y-mu_ymu_x+mu_xmu_y
(9)
=<xy>-<x><y>,
(10)

where mu_i is the mean for the variable i.


See also

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

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