### 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−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 probability density function with parameter λ at x, namely e−λλk/k!.

#### Example

poisson(10.0,9)

0.125110035721 |  |  |  |  |  |  |  |  |  |

##### 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

poisson_cdf(10.0,3)

0.0103360506759 |  |  |  |  |  |  |  |  |  |

poisson_cdf(10.0,3,10)

0.580270354477 |  |  |  |  |  |  |  |  |  |

##### 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

poisson_icdf(10.0,0.975)

17 |  |  |  |  |  |  |  |  |  |