# The Poisson distribution

### 8.4.6 The Poisson distribution

#### The probability density function for the Poisson distribution: poisson

Recall that for the Poisson distribution with parameter λ, the probability of a non-negative integer k is e−λλk/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 density function with parameter λ at x, namely e−λλk/k!.

#### Example.

Input:

poisson(10.0,9)

Output:

0.125110035721

#### The cumulative distribution function for the Poisson distribution: poisson_cdf

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) + … + binomial(µ,floor(x))

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

  Prob(x ≤ X ≤ y) = poisson(µ,ceil(x)) + … + poisson(µ,floor(y))

#### Examples.

- Input:
  poisson_cdf(10.0,3)
  Output:

  0.0103360506759

- Input:
  poisson_cdf(10.0,3,10)
  Output:

  0.580270354477

#### The inverse distribution function for the Poisson distribution: poisson_icdf

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.

Input:

poisson_icdf(10.0,0.975)

Output:

17