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Expectation Value

The expectation value, also called the expected value, mathematical expectation, or mean, of a random variable is its probability-weighted average. No single notation is universal. Common choices with an ordinary italic E include E{X} (Papoulis 1984), E(X) (JCGM 2011, §3.16), E[X] (Kantor et al. 2015, p. 35), and EX (Durrett 2019, p. 29). An upright operator E[X] is another convention (Löffler and Kruschwitz 2019), while the blackboard-bold form E[X], angle brackets <X>, and the symbols mu_X and mu are also used (Bauckhage and Sifa 2026).

In this work, the notation E[X] is used. In the formulas below, the random variables are real-valued, the functions are measurable and real-valued, and all displayed expectations are finite.

For a function f of a discrete random variable X, the expectation value is

E[f(X)]=sum_(x)f(x)P(X=x).
(1)

For a function f of a continuous random variable X with probability density function p_X, it is

E[f(X)]=int_(-infty)^inftyf(x)p_X(x)dx.
(2)

The expectation is a linear operator, so

E[aX+bY]=aE[X]+bE[Y]
(3)
E[a]=a
(4)
E[sum_(i=1)^(n)X_i]=sum_(i=1)^(n)E[X_i],
(5)

where X, Y, and the X_i are random variables and a and b are real constants.

For discrete random variables X_1, X_2, ..., X_n, the expectation of a function f(X_1,X_2,...,X_n) is

E[f(X_1,X_2,...,X_n)]=sum_(x_1,x_2,...,x_n)f(x_1,x_2,...,x_n)P(X_1=x_1,X_2=x_2,...,X_n=x_n).
(6)

For continuous random variables with joint probability density function p_(X_1,X_2,...,X_n),

E[f(X_1,X_2,...,X_n)]=int...intf(x_1,x_2,...,x_n)p_(X_1,X_2,...,X_n)(x_1,x_2,...,x_n)dx_1...dx_n.
(7)

In particular, the covariance of two random variables X and Y is

E[(X-mu_X)(Y-mu_Y)]=E[XY]-E[X]E[Y],
(8)

where mu_X=E[X] and mu_Y=E[Y].

See also

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

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References

Bauckhage, C. and Sifa, R. "Boolean Domains, Numbers, and Vectors." Ch. 2 in Quantum Computing from Hopfield Nets. Cham, Switzerland: Springer, pp. 29-54, 2026. https://doi.org/10.1007/978-3-031-99402-9_2.Durrett, R. Probability: Theory and Examples, 5th ed. Cambridge, England: Cambridge University Press, 2019.Joint Committee for Guides in Metrology. "Evaluation of Measurement Data-Supplement 2 to the Guide to the Expression of Uncertainty in Measurement-Extension to Any Number of Output Quantities." JCGM 102:2011. https://doi.org/10.59161/JCGM102-2011.Kantor, I.; Matoušek, J.; and Šámal, R. Mathematics++: Selected Topics Beyond the Basic Courses. Providence, RI: American Mathematical Society, p. 35, 2015.Löffler, A. and Kruschwitz, L. "Expectation and Lebesgue Integral." Ch. 5 in The Brownian Motion: A Rigorous but Gentle Introduction for Economists. Cham, Switzerland: Springer, 2019. https://doi.org/10.1007/978-3-030-20103-6_5.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.

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Expectation Value

Cite this as:

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

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