ZIPoisson {distributions3} R Documentation

## Create a zero-inflated Poisson distribution

### Description

Zero-inflated Poisson distributions are frequently used to model counts with many zero observations.

### Usage

ZIPoisson(lambda, pi)


### Arguments

 lambda Parameter of the Poisson component of the distribution. Can be any positive number. pi Zero-inflation probability, can be any value in ⁠[0, 1]⁠.

### Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a zero-inflated Poisson random variable with parameter lambda = \lambda.

Support: \{0, 1, 2, 3, ...\}

Mean: (1 - \pi) \cdot \lambda

Variance: (1 - \pi) \cdot \lambda \cdot (1 + \pi \cdot \lambda)

Probability mass function (p.m.f.):

 P(X = k) = \pi \cdot I_{0}(k) + (1 - \pi) \cdot f(k; \lambda) 

where I_{0}(k) is the indicator function for zero and f(k; \lambda) is the p.m.f. of the Poisson distribution.

Cumulative distribution function (c.d.f.):

 P(X \le k) = \pi + (1 - \pi) \cdot F(k; \lambda) 

where F(k; \lambda) is the c.d.f. of the Poisson distribution.

Moment generating function (m.g.f.):

 E(e^{tX}) = \pi + (1 - \pi) \cdot e^{\lambda (e^t - 1)} 

### Value

A ZIPoisson object.

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), ZINegativeBinomial(), ZTNegativeBinomial(), ZTPoisson()

### Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)


[Package distributions3 version 0.2.1 Index]