Price's Theorem

Consider a bivariate normal distribution in variables  and  with covariance

(1)

and an arbitrary function . Then the expected value of the random variable 

(2)

satisfies

(3)

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.