The q-Bernstein basis as a q-binomial distribution

Abstract

The q-Bernstein basis, used in the definition of the q-Bernstein polynomials, is shown to be the probability mass function of a q-binomial distribution. This distribution is defined on a sequence of zero–one Bernoulli trials with probability of failure at any trial increasing geometrically with the number of previous failures. A modification of this model, with the probability of failure at any trial decreasing geometrically with the number of previous failures, leads to a second q-binomial distribution that is also connected to the q-Bernstein polynomials. The q-factorial moments as well as the usual factorial moments of these distributions are derived. Further, the q-Bernstein polynomial B n ( f( t), q; x) is expressed as the expected value of the function f([ X n ] q /[ n] q ) of the random variable X n obeying the q-binomial distribution. Also, using the expression of the q-moments of X n , an explicit expression of the q-Bernstein polynomial B n ( f r ( t), q; x), for f r ( t) a polynomial, is obtained.