ZTPoisson: Create a zero-truncated Poisson distribution

In distributions3: Probability Distributions as S3 Objects

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.

See Also

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

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.