The q-Bernstein basis as a q-binomial distribution

作者：Charalambides Charalambos A^*

来源：Journal of Statistical Planning and Inference, 2010, 140(8): 2184-2190.

DOI：10.1016/j.jspi.2010.01.014

• q-Bernstein polynomials
• q-Distributions
• q-Stirling numbers

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.

• 出版日期2010-8