dist_gamma
The Gamma distribution
Description
![[Stable]](../help/figures/lifecycle-stable.svg)
Usage
dist_gamma(shape, rate)
Arguments
shape |
shape and scale parameters. Must be positive,
|
rate |
an alternative way to specify the scale. |
Details
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.
We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.
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
See Also
Examples
dist <- dist_gamma(shape = c(1,2,3,5,9,7.5,0.5), rate = c(0.5,0.5,0.5,1,2,1,1)) dist mean(dist) variance(dist) skewness(dist) kurtosis(dist) generate(dist, 10) density(dist, 2) density(dist, 2, log = TRUE) cdf(dist, 4) quantile(dist, 0.7)