Binomial distribution and complemented binomial distribution
The binomial distribution is defined as the sum of the terms 0 through k
of the Binomial probability density.
The complement returns the sum of the terms k+1 through n.
binomialDistribution = $(BIGSUM j=0, k) $(CHOOSE n, j) pj(1-p)n-j
binomialDistributionCompl = $(BIGSUM j=k+1, n) $(CHOOSE n, j) pj(1-p)n-j
The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula
y = binomialDistribution( k, n, p ) = betaDistribution( n-k, k+1, 1-p ).
The arguments must be positive, with p ranging from 0 to 1, and k<=n.
Binomial distribution and complemented binomial distribution
The binomial distribution is defined as the sum of the terms 0 through k of the Binomial probability density. The complement returns the sum of the terms k+1 through n.
binomialDistribution = $(BIGSUM j=0, k) $(CHOOSE n, j) pj (1-p)n-j
binomialDistributionCompl = $(BIGSUM j=k+1, n) $(CHOOSE n, j) pj (1-p)n-j
The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula
y = binomialDistribution( k, n, p ) = betaDistribution( n-k, k+1, 1-p ).
The arguments must be positive, with p ranging from 0 to 1, and k<=n.