| 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 |
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.
See Also
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)