20.4.6  Poisson distribution

The probability density function for the Poisson distribution.

Recall that for the Poisson distribution with parameter λ, the probability of a non-negative integer k is e^−λλ^k/k!. This distribution has mean λ and variance λ.

The poisson command gives the density function for the Poisson distribution.

• poisson takes two arguments:
  • λ, a real number.
  • k, a non-negative integer.
• poisson(λ,k) returns the value of the Poisson probability density function with parameter λ at x, namely e^−λλ^k/k!.

Example

The cumulative distribution function for the Poisson distribution.

The poisson_cdf command computes the cumulative distribution function for the Poisson distribution.

• poisson_cdf takes two arguments:
  • µ, a real number.
  • x, a real number.
  • Optionally, y, a real number.
• poisson_cdf(µ,x) returns

  Prob(X ≤ x)=poisson(µ,0)+⋯+ poisson(µ,⌊ x⌋))

  for the Poisson distribution with parameter µ.
• poisson_cdf(µ, x, y) returns

  Prob(x ≤ X ≤ y)=poisson(µ,⌈ x⌉)+⋯+ poisson(µ,⌊ y⌋)).

Examples

The inverse distribution function for the Poisson distribution.

The poisson_icdf command finds the inverse distribution function for the Poisson distribution.

• poisson_icdf takes three arguments:
  • µ, a real number.
  • h, a real number between 0 and 1.
• poisson_icdf(µ,h) returns the value of the inverse distribution for the Poisson distribution with parameter µ; namely, the value of x for which Prob(X ≤ x)=h.

Example