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Price's Theorem

Consider a bivariate normal distribution in variables x and y with covariance

rho=rho_(11)=<xy>-<x><y>
(1)

and an arbitrary function g(x,y). Then the expected value of the random variable g(x,y)

<g(x,y)>=int_(-infty)^inftyint_(-infty)^inftyg(x,y)f(x,y)dxdy
(2)

satisfies

(partial^n<g(x,y)>)/(partialrho^n)=<(partial^(2n)g(x,y))/(partialx^npartialy^n)>.
(3)

See also

Covariance, Bivariate Normal Distribution

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References

McMahon, E. L. "An Extension of Price's Theorem." IEEE Trans. Inform. Th. 10, 168-171, 1964.Papoulis, A. "Price's Theorem and Join Moments." Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 226-228, 1984.Price, R. "A Useful Theorem for Non-Linear Devices Having Gaussian Inputs." IEEE Trans. Inform. Th. 4, 69-72, 1958.

Referenced on Wolfram|Alpha

Price's Theorem

Cite this as:

Weisstein, Eric W. "Price's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PricesTheorem.html

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