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 polynomials

A family of polynomials Pk is orthogonal over the interval [-1, 1] with respect to a weight w(x) if

\int_{-1}^1 P_m(x) P_n(x) w(x) \, dx = 0

whenever mn.

If w(x) = 1, we get the Legendre polynomials.

If w(x) = (1 – x²)-1/2 we get the Chebyshev polynomials.

These are both special cases of the Jacobi polynomials which have weight w(x) = (1- x)α (1 + x)β. Legendre polynomials correspond to α = β = 0, and Chebyshev polynomials correspond to α = β = -1/2.

Connection to beta distribution

The weight function for Jacobi polynomials is a rescaling of the density function of a beta distribution. The change of variables x = 1 – 2u shows

\int_{-1}^1 f(x) (1-x)^\alpha (1+x)^\beta \, dx = 2^{\alpha + \beta + 1}\int_0^1 f(1-2u) u^\alpha (1-u)^\beta \,du

The right side is proportional to the expected value of f(1 – 2X) where X is a random variable with a beta(α + 1, β+1) distribution. So for fixed α and β, if mn and X has a beta(α + 1, β+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 polynomials

The Laguerre polynomials are orthogonal over the interval [0, ∞) with weight w(x) = xα exp(-x), which is proportional to the density of a gamma random variable with shape α+1 and scale 1.

Hermite polynomials

There are two minor variations on the Hermite polynomials, depending on whether you take the weight to be exp(-x²) or exp(-x²/2). 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 (-∞, ∞) with the standard normal (Gaussian) density as the weight.

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