ZTPoisson | R Documentation |
Zero-truncated Poisson distributions are frequently used to model counts where zero observations cannot occur or have been excluded.
ZTPoisson(lambda)
lambda |
Parameter of the underlying untruncated Poisson distribution. Can be any positive number. |
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
= λ.
Support: {1, 2, 3, ...}
Mean:
λ \cdot 1/(1 - e^{-λ})
Variance: m \cdot (λ + 1 - m), where m is the mean above.
Probability mass function (p.m.f.):
P(X = k) = f(k; λ)/(1 - f(0; λ))
where f(k; λ) is the p.m.f. of the Poisson
distribution.
Cumulative distribution function (c.d.f.):
P(X = k) = F(k; λ)/(1 - F(0; λ))
where F(k; λ) is the c.d.f. of the Poisson
distribution.
Moment generating function (m.g.f.):
E(e^(tX)) = 1/(1 - e^{-λ}) \cdot e^(λ (e^t - 1))
A ZTPoisson
object.
Other discrete distributions:
Bernoulli()
,
Binomial()
,
Categorical()
,
Geometric()
,
HurdleNegativeBinomial()
,
HurdlePoisson()
,
HyperGeometric()
,
Multinomial()
,
NegativeBinomial()
,
Poisson()
,
ZINegativeBinomial()
,
ZIPoisson()
,
ZTNegativeBinomial()
## 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)
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