dist_gamma {distributional} | 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.
dist_gamma(shape, rate, scale = 1/rate)
shape, scale |
shape and scale parameters. Must be positive,
|
rate |
an alternative way to specify the scale. |
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
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)