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Binomial Distributions Are Bernstein Vectors

This Demonstration illustrates the intrinsic connection between binomial distributions and Bernstein polynomials. The Bernstein polynomial of degree and index , , is equal to the probability of observing hits in identical draws with probability of a hit on each draw.
You can set the degree or the index to be between 0 and .
You can view graphs of Bernstein polynomials for various indices as slices across the hits axis or binomial distributions for various probability values as slices across the probability axis.
Coarsening the probability grid gives shorter response times. In addition, you can overlay the respective continuous distributions.

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DETAILS

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 .
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