# Pois-class

From distr v2.6
by Peter Ruckdeschel

##### Class "Pois"

The Poisson distribution has density

$$p(x) = \frac{\lambda^x e^{-\lambda}}{x!}$$ for $x = 0, 1, 2, \ldots$. The mean and variance are $E(X) = Var(X) = \lambda$.

C.f. `rpois`

- Keywords
- distribution

##### Note

Working with a computer, we use a finite interval as support which carries at least mass `1-getdistrOption("TruncQuantile")`

.

##### Objects from the Class

Objects can be created by calls of the form `Pois(lambda)`

.
This object is a Poisson distribution.

##### Slots

`img`

- Object of class
`"Naturals"`

: The space of the image of this distribution has got dimension 1 and the name "Natural Space". `param`

- Object of class
`"PoisParameter"`

: the parameter of this distribution (lambda), declared at its instantiation `r`

- Object of class
`"function"`

: generates random numbers (calls function rpois) `d`

- Object of class
`"function"`

: density function (calls function dpois) `p`

- Object of class
`"function"`

: cumulative function (calls function ppois) `q`

- Object of class
`"function"`

: inverse of the cumulative function (calls function qpois). The quantile is defined as the smallest value $x$ such that $F(x) \ge p$, where $F$ is the distribution function. `support`

- Object of class
`"numeric"`

: a (sorted) vector containing the support of the discrete density function `.withArith`

- logical: used internally to issue warnings as to interpretation of arithmetics
`.withSim`

- logical: used internally to issue warnings as to accuracy
`.logExact`

- logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function
`.lowerExact`

- logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function
`Symmetry`

- object of class
`"DistributionSymmetry"`

; used internally to avoid unnecessary calculations.

##### Extends

Class `"DiscreteDistribution"`

, directly.
Class `"UnivariateDistribution"`

, by class `"DiscreteDistribution"`

.
Class `"Distribution"`

, by class `"DiscreteDistribution"`

.

##### Methods

- +
`signature(e1 = "Pois", e2 = "Pois")`

: For the Poisson distribution the exact convolution formula is implemented thereby improving the general numerical approximation.- initialize
`signature(.Object = "Pois")`

: initialize method- lambda
`signature(object = "Pois")`

: returns the slot lambda of the parameter of the distribution- lambda<-
`signature(object = "Pois")`

: modifies the slot lambda of the parameter of the distribution

##### See Also

`PoisParameter-class`

`DiscreteDistribution-class`

`Naturals-class`

`rpois`

##### Examples

```
P <- Pois(lambda = 1) # P is a Poisson distribution with lambda = 1.
r(P)(1) # one random number generated from this distribution, e.g. 1
d(P)(1) # Density of this distribution is 0.3678794 for x = 1.
p(P)(0.4) # Probability that x < 0.4 is 0.3678794.
q(P)(.1) # x = 0 is the smallest value x such that p(B)(x) >= 0.1.
lambda(P) # lambda of this distribution is 1.
lambda(P) <- 2 # lambda of this distribution is now 2.
R <- Pois(lambda = 3) # R is a Poisson distribution with lambda = 2.
S <- P + R # R is a Poisson distribution with lambda = 5(=2+3).
```

*Documentation reproduced from package distr, version 2.6, License: LGPL-3*

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