Binomial Distributions Are Bernstein Vectors
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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.
Contributed by: Ludwig Weingarten (February 2010)
Open content licensed under CC BY-NC-SA
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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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