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

.