20.4.3  Binomial distribution

The probability density function for the binomial distribution.

If you perform an experiment n times where the probability of success each time is p, then the probability of exactly k successes is:

This determines the binomial distribution.

The binomial command computes the density function for the binomial distribution.

• binomial takes two mandatory arguments and one optional argument.
  • n, a positive integer.
  • k, a nonnegative integer less than or equal to n.
  • Optionally, p, a probability (a real number between 0 and 1).
• binomial(n,k) returns the binomial coefficient (_kⁿ) (see Section 12.1.2), same as comb(n,k).
• binomial(n,k,p) returns the probability given by (1).

Examples

or:

The cumulative distribution function for the binomial distribution.

The binomial_cdf command computes the cumulative distribution function for the binomial distribution.

• binomial_cdf takes three mandatory arguments and one optional argument:
  • n, a positive integer.
  • p, a probability (a real number between 0 and 1).
  • x, a real number.
  • Optionally, y, a real number.
• binomial_cdf(n,p,x) returns

  Prob(X ≤ x)=binomial(n,0,p)+⋯+ binomial(n,⌊ x⌋),p)

• binomial_cdf(n,p,x,y) returns

  Prob(x ≤ X ≤ y)=binomial(n,⌈ x⌉,p)+⋯+ binomial(n,⌊ y⌋,p)

Examples

The inverse distribution function for the binomial distribution.

The binomial_icdf command computes the inverse distribution function for the binomial distribution.

• binomial_icdf takes three mandatory arguments and one optional argument:
  • n, a positive integer.
  • p, a probability (a real number between 0 and 1).
  • h, a real number between 0 and 1.
• binomial_icdf(n,p,h) returns the value of the inverse distribution for the binomial distribution with n trials and probability p; namely, the smallest value of x for which Prob(X ≤ x) ≥ h.

Example

Note that binomial_cdf(4,0.5,3)=0.9375, which is bigger than 0.9, while binomial_cdf(4,0.5,2)=0.6875, which is smaller than 0.9.