ZIPoisson | R Documentation |
Zero-inflated Poisson distributions are frequently used to model counts with many zero observations.
ZIPoisson(lambda, pi)
lambda |
Parameter of the Poisson component of the distribution. Can be any positive number. |
pi |
Zero-inflation probability, can be any value in |
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
= λ.
Support: {0, 1, 2, 3, ...}
Mean: (1 - π) \cdot λ
Variance: (1 - π) \cdot λ \cdot (1 + π \cdot λ)
Probability mass function (p.m.f.):
P(X = k) = π \cdot I_{0}(k) + (1 - π) \cdot f(k; λ)
where I_{0}(k) is the indicator function for zero and
f(k; λ) is the p.m.f. of the Poisson
distribution.
Cumulative distribution function (c.d.f.):
P(X ≤ k) = π + (1 - π) \cdot F(k; λ)
where F(k; λ) is the c.d.f. of the Poisson
distribution.
Moment generating function (m.g.f.):
E(e^(tX)) = π + (1 - π) \cdot e^(λ (e^t - 1))
A ZIPoisson
object.
Other discrete distributions:
Bernoulli()
,
Binomial()
,
Categorical()
,
Geometric()
,
HurdleNegativeBinomial()
,
HurdlePoisson()
,
HyperGeometric()
,
Multinomial()
,
NegativeBinomial()
,
Poisson()
,
ZINegativeBinomial()
,
ZTNegativeBinomial()
,
ZTPoisson()
## 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)
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