The Poisson distribution

### 8.4.4 The Poisson distribution

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

Recall that for the Poisson distribution with parameter µ, the probability of a non-negative integer k is e−µµk/k!. It will mean µ and variance µ. The poisson command will find this value, given µ and k. For example,

poisson(10.0,9)

is

0.125110035721

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

The cumulative distribution function for the Poisson distribution is given by the poisson_cdf command with arguments µ and x; poisson_cdf(µ,x) = Prob(X ≤ x). If you enter

poisson_cdf(10.0,3)

you will get

0.0103360506759

With another argument, poisson_cdf will find the probability of falling between two values; poisson_cdf(µ,x,y) = Prob(x ≤ X ≤ y). If you enter

poisson_cdf(10.0,3,10)

you will get

0.580270354477

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

Given a value h, the inverse distribution function gives the smallest value of x so that Prob(X ≤ x) ≥ h. Given arguments of a parameter µ and a value x, the poisson_icdf gives the inverse distribution function for the poisson distribution. If you enter

poisson_icdf(10.0,0.975)

you will get

17