# Pois-class

##### 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

```
# NOT RUN {
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.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
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 2.7.0, License: LGPL-3*