Binomial Distributions Are Bernstein Vectors

A binomial distribution (in Mathematica, the built-in function BinomialDistribution[,]), for single-draw probability and a maximum number of draws , is a vector with entries (counting from 0 to ).

The component of this vector gives the probability of having exactly hits in draws when a single draw has the probability to hit.
This is known to be , so the distribution can be seen as a family of vectors indexed by .
For more information, see T. H. Wonnacott and R. J. Wonnacott, Introductory Statistics, 4th ed., New York: Wiley, 1985.

Bernstein polynomials, , are weighted multiples of and of the form , where is the degree, is the index running from 0 to , and . So, for each degree , there are polynomial functions from to . The only zeros of these functions are 0 and 1; the index counts the multiplicity of the root at 0 and counts the multiplicity of the root at 1. In addition, they are positive in , nonnegative in [0,1] and, for each , they sum to the constant function 1 on , so they constitute a partition of unity.

In Mathematica these polynomials are denoted as BernsteinBasis[d,i,t].

Thinking of the variable as a probability , a Bernstein vector of degree and probability can thus be defined as , a vector of functions from to with entries.

So, for each fixed probability , this -dimensional Bernstein vector equals the binomial distribution .