| Poisson {distributions3} | R Documentation |
Create a Poisson distribution
Description
Poisson distributions are frequently used to model counts.
Usage
Poisson(lambda)
Arguments
lambda |
The shape parameter, which is also the mean and the variance of the 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 Poisson random variable with parameter
lambda = \lambda.
Support: \{0, 1, 2, 3, ...\}
Mean: \lambda
Variance: \lambda
Probability mass function (p.m.f):
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
Cumulative distribution function (c.d.f):
P(X \le k) = e^{-\lambda}
\sum_{i = 0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!}
Moment generating function (m.g.f):
E(e^{tX}) = e^{\lambda (e^t - 1)}
Value
A Poisson object.
See Also
Other discrete distributions:
Bernoulli(),
Binomial(),
Categorical(),
Geometric(),
HurdleNegativeBinomial(),
HurdlePoisson(),
HyperGeometric(),
Multinomial(),
NegativeBinomial(),
ZINegativeBinomial(),
ZIPoisson(),
ZTNegativeBinomial(),
ZTPoisson()
Examples
set.seed(27)
X <- Poisson(2)
X
random(X, 10)
pdf(X, 2)
log_pdf(X, 2)
cdf(X, 4)
quantile(X, 0.7)
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))