| 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.
See Also
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)