18.4.6 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
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⌋))
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for the Poisson distribution with parameter µ.
- poisson_cdf(µ, x, y) returns
Prob(x ≤ X ≤ y)=poisson(µ,⌈ x⌉)+⋯+
poisson(µ,⌊ y⌋)).
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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