20.4.13  Geometric distribution

The probability density function for the geometric distribution.

If an experiment with probability of success p is iterated, the probability that the first success occurs on the kth trial is (1−p)^k−1p. This gives the geometric distribution (with parameter p) on the natural numbers. Given such a p, the geometric density function at n is given by

The geometric command computes this density function.

• geometric takes two arguments:
  • p, a probability (a number between 0 and 1).
  • x, a real number.
  geometric(p,x) returns the value of the geometric density function with probability p, given in (10).

Example

The cumulative distribution function of the geometric distribution.

The geometric_cdf command computes the cumulative distribution function for the geometric distribution.

• geometric_cdf takes three mandatory arguments and one optional argument:
  • p, a probability (a number between 0 and 1).
  • n, a natural number.
  • Optionally, k, a natural number.
• betad_cdf(p,n) returns Prob(X ≤ n) for the geometric distribution with probability p.
• beta_cdf(p,n,k) returns Prob(n ≤ X ≤ k).

It turns out that geometric_cdf(p,n)=1−(1−p)ⁿ.

Examples

The inverse distribution function for the geometric distribution.

The geometric_icdf command computes the inverse distribution for the geometric distribution.

• geometric_icdf takes two arguments:
  • p, a probability (a number between 0 and 1).
  • h, a real number between 0 and 1.
• geometric_icdf(a,b,h) returns the inverse distribution for the geometric distribution with probability p; namely, the smallest natural number n for which Prob(X ≤ n) ≥ h.

Example