Orthogonal polynomials and the beta distribution
This post shows a connection between three families of orthogonal polynomials-Legendre, Chebyshev, and Jacobi-and the beta distribution.
Legendre, Chebyshev, and Jacobi polynomialsA family of polynomials Pk is orthogonal over the interval [-1, 1] with respect to a weight w(x) if
whenever m a n.
If w(x) = 1, we get the Legendre polynomials.
If w(x) = (1 - x^2)-1/2 we get the Chebyshev polynomials.
These are both special cases of the Jacobi polynomials which have weight w(x) = (1- x)I (1 + x)I^2. Legendre polynomials correspond to I = I^2 = 0, and Chebyshev polynomials correspond to I = I^2 = -1/2.
Connection to beta distributionThe weight function for Jacobi polynomials is a rescaling of the density function of a beta distribution. The change of variables x = 1 - 2u shows
The right side is proportional to the expected value of f(1 - 2X) where X is a random variable with a beta(I + 1, I^2+1) distribution. So for fixed I and I^2, if m a n and X has a beta(I + 1, I^2+1) distribution, then the expected value of Pm(1 - 2X) Pn(1 - 2X) is zero.
While we're at it, we'll briefly mention two other connections between orthogonal polynomials and probability: Laguerre polynomials and Hermite polynomials.
Laguerre polynomialsThe Laguerre polynomials are orthogonal over the interval [0, a) with weight w(x) = xI exp(-x), which is proportional to the density of a gamma random variable with shape I+1 and scale 1.
Hermite polynomialsThere are two minor variations on the Hermite polynomials, depending on whether you take the weight to be exp(-x^2) or exp(-x^2/2). These are sometimes known as the physicist's Hermite polynomials and the probabilist's Hermite polynomials. Naturally we're interested in the latter. The probabilist's Hermite polynomials are orthogonal over (-a, a) with the standard normal (Gaussian) density as the weight.
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