9.4.6 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:
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:
- Input:
poisson_cdf(10.0,3,10)
Output:
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: