Pois: Poisson Distribution

In joker: Probability Distributions and Parameter Estimation

Poisson Distribution

Description

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that the events occur with a constant rate \lambda > 0 and independently of the time since the last event.

Usage

Pois(lambda = 1)

## S4 method for signature 'Pois,numeric'
d(distr, x, log = FALSE)

## S4 method for signature 'Pois,numeric'
p(distr, q, lower.tail = TRUE, log.p = FALSE)

## S4 method for signature 'Pois,numeric'
qn(distr, p, lower.tail = TRUE, log.p = FALSE)

## S4 method for signature 'Pois,numeric'
r(distr, n)

## S4 method for signature 'Pois'
mean(x)

## S4 method for signature 'Pois'
median(x)

## S4 method for signature 'Pois'
mode(x)

## S4 method for signature 'Pois'
var(x)

## S4 method for signature 'Pois'
sd(x)

## S4 method for signature 'Pois'
skew(x)

## S4 method for signature 'Pois'
kurt(x)

## S4 method for signature 'Pois'
entro(x)

## S4 method for signature 'Pois'
finf(x)

llpois(x, lambda)

## S4 method for signature 'Pois,numeric'
ll(distr, x)

epois(x, type = "mle", ...)

## S4 method for signature 'Pois,numeric'
mle(distr, x, na.rm = FALSE)

## S4 method for signature 'Pois,numeric'
me(distr, x, na.rm = FALSE)

vpois(lambda, type = "mle")

## S4 method for signature 'Pois'
avar_mle(distr)

## S4 method for signature 'Pois'
avar_me(distr)

Arguments

lambda	numeric. The distribution parameter.
distr	an object of class Pois.
x	For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Pois. For the log-likelihood and the estimation functions, x is the sample of observations.
log, log.p	logical. Should the logarithm of the probability be returned?
q	numeric. Vector of quantiles.
lower.tail	logical. If TRUE (default), probabilities are P(X \leq x), otherwise P(X > x).
p	numeric. Vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
type	character, case ignored. The estimator type (mle or me).
...	extra arguments.
na.rm	logical. Should the NA values be removed?

Details

The probability mass function (PMF) of the Poisson distribution is:

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k \in \mathbb{N}_0.

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dpois(), ppois(), qpois(), rpois()

Examples

# -----------------------------------------------------
# Pois Distribution Example
# -----------------------------------------------------

# Create the distribution
lambda <- 5
D <- Pois(lambda)

# ------------------
# dpqr Functions
# ------------------

d(D, 0:10) # density function
p(D, 0:10) # distribution function
qn(D, c(0.4, 0.8)) # inverse distribution function
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llpois(x, lambda)

epois(x, type = "mle")
epois(x, type = "me")

mle(D, x)
me(D, x)
e(D, x, type = "mle")

mle("pois", x) # the distr argument can be a character

# ------------------
# Estimator Variance
# ------------------

vpois(lambda, type = "mle")
vpois(lambda, type = "me")

avar_mle(D)
avar_me(D)

v(D, type = "mle")

joker documentation built on June 8, 2025, 12:12 p.m.