Gamma | R Documentation |
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/β. 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(α_1, β) and
X_2 is Gamma(α_2, β), a function of these two variables
of the form \frac{X_1}{X_1 + X_2} Beta(α_1, α_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.
Gamma(shape, rate = 1)
shape |
The shape parameter. Can be any positive number. |
rate |
The rate parameter. Can be any positive number. Defaults
to |
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
= α and
rate
= β.
Support: x \in (0, ∞)
Mean: \frac{α}{β}
Variance: \frac{α}{β^2}
Probability density function (p.m.f):
f(x) = \frac{β^{α}}{Γ(α)} x^{α - 1} e^{-β x}
Cumulative distribution function (c.d.f):
f(x) = \frac{Γ(α, β x)}{Γ{α}}
Moment generating function (m.g.f):
E(e^(tX)) = \Big(\frac{β}{ β - t}\Big)^{α}, \thinspace t < β
A Gamma
object.
Other continuous distributions:
Beta()
,
Cauchy()
,
ChiSquare()
,
Erlang()
,
Exponential()
,
FisherF()
,
Frechet()
,
GEV()
,
GP()
,
Gumbel()
,
LogNormal()
,
Logistic()
,
Normal()
,
RevWeibull()
,
StudentsT()
,
Tukey()
,
Uniform()
,
Weibull()
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))
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