ZTPoisson {distributions3} R Documentation

## Create a zero-truncated Poisson distribution

### Description

Zero-truncated Poisson distributions are frequently used to model counts where zero observations cannot occur or have been excluded.

### Usage

ZTPoisson(lambda)


### Arguments

 lambda Parameter of the underlying untruncated Poisson distribution. Can be any positive number.

### 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-truncated Poisson random variable with parameter lambda = \lambda.

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

Mean:

 \lambda \cdot \frac{1}{1 - e^{-\lambda}} 

Variance: m \cdot (\lambda + 1 - m), where m is the mean above.

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

 P(X = k) = \frac{f(k; \lambda)}{1 - f(0; \lambda)} 

where f(k; \lambda) is the p.m.f. of the Poisson distribution.

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

 P(X = k) = \frac{F(k; \lambda)}{1 - F(0; \lambda)} 

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

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

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

### Value

A ZTPoisson object.

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

### Examples

## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
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]