## Article

#### Title: On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials

##### Issue: Volume 82 Series A Part 1 Year 2020
###### Abstract
Consider a two-dimensional random vector (X, Y )T. Let Q0,Q1, . . . denote orthogonal polynomials with respect to the marginal distribution of X and let P0, P1, . . . denote orthogonal polynomials with respect to the marginal distribution of Y . In this paper, identities of the form E[Pn(Y ) |X] = anQn(X), for constants a0, a1, . . . are considered and necessary and sufficient conditions for this type of identity to hold are given,. The application of the identity to the maximal correlation of two random variables and to the L2 completeness ofa bivariate distribution are discussed.