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. The top control sets . One can view binomial distributions for various -values as slices across the probability axis and graphs of Bernstein polynomials as slices across the hits axis.
Coarsening the probability grid gives shorter response times.
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 they are 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.
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 .
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