Orthogonal polynomials via random variables
Abstract
Every set of orthonormal polynomials, {fn(x)}, is generated by an associated measure. This can be scaled to be the distribution of a random variable X with finite moments. For example, uniform, normal, exponential, gamma, beta, Poisson and Binomial random variables generate the Legendre, Hermite, Laguerre, Jacobi, ultraspherical, Poisson-Charlier and Krawtchouk polynomials. One advantage of this viewpoint is the simplification made when using central moments or cumulants rather than raw moments. Also {f n(X)} are uncorrelated random variables. We compare three ways of generating orthonormal polynomials from a random variable, and argue that the best way is a particular central version of the Gram-Schmidt procedure. We give explicit formulas for the first six polynomials generated by a random variable, as well as recurrence formulas for the general polynomial.
