| Gamma {distributions3} | R Documentation |
Create a Gamma distribution
Description
Several important distributions are special cases of the Gamma
distribution. When the shape parameter is 1, the Gamma is an
exponential distribution with parameter 1/\beta. When the
shape = n/2 and rate = 1/2, the Gamma is a equivalent to
a chi squared distribution with n degrees of freedom. Moreover, if
we have X_1 is Gamma(\alpha_1, \beta) and
X_2 is Gamma(\alpha_2, \beta), a function of these two variables
of the form \frac{X_1}{X_1 + X_2} Beta(\alpha_1, \alpha_2).
This last property frequently appears in another distributions, and it
has extensively been used in multivariate methods. More about the Gamma
distribution will be added soon.
Usage
Gamma(shape, rate = 1)
Arguments
shape |
The shape parameter. Can be any positive number. |
rate |
The rate parameter. Can be any positive number. Defaults
to |
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 Gamma random variable
with parameters
shape = \alpha and
rate = \beta.
Support: x \in (0, \infty)
Mean: \frac{\alpha}{\beta}
Variance: \frac{\alpha}{\beta^2}
Probability density function (p.m.f):
f(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}
Cumulative distribution function (c.d.f):
f(x) = \frac{\Gamma(\alpha, \beta x)}{\Gamma{\alpha}}
Moment generating function (m.g.f):
E(e^{tX}) = \Big(\frac{\beta}{ \beta - t}\Big)^{\alpha}, \thinspace t < \beta
Value
A Gamma object.
See Also
Other continuous distributions:
Beta(),
Cauchy(),
ChiSquare(),
Erlang(),
Exponential(),
FisherF(),
Frechet(),
GEV(),
GP(),
Gumbel(),
LogNormal(),
Logistic(),
Normal(),
RevWeibull(),
StudentsT(),
Tukey(),
Uniform(),
Weibull()
Examples
set.seed(27)
X <- Gamma(5, 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))