There are advantages in viewing orthogonal functions as functions generated by a random variable from a basis set of functions. Let Y be a random variable distributed uniformly on [0,1]. We give two ways of generating the Zernike radial polynomials with parameter l, {Zll+2n(x), n≥0}. The first is using the standard basis {xn,n≥0} and the random variable Y1/(l+1). The second is using the nonstandard basis {xl+2n,n≥0} and the random variable Y1/2. Zernike polynomials are important in the removal of lens aberrations, in characterizing video images with a small number of numbers, and in automatic aircraft identification.
