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.
dist_gamma(shape, rate, scale = 1/rate)
shape and scale parameters. Must be positive,
scale
strictly.
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
= \(\alpha\) 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):
$$ E(e^{tX}) = \Big(\frac{\beta}{ \beta - t}\Big)^{\alpha}, \thinspace t < \beta $$
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
#> <distribution[7]>
#> [1] Γ(1, 0.5) Γ(2, 0.5) Γ(3, 0.5) Γ(5, 1) Γ(9, 2) Γ(7.5, 1) Γ(0.5, 1)
mean(dist)
#> [1] 2.0 4.0 6.0 5.0 4.5 7.5 0.5
variance(dist)
#> [1] 4.00 8.00 12.00 5.00 2.25 7.50 0.50
skewness(dist)
#> [1] 2.0000000 1.4142136 1.1547005 0.8944272 0.6666667 0.7302967 2.8284271
kurtosis(dist)
#> [1] 6.0000000 3.0000000 2.0000000 1.2000000 0.6666667 0.8000000 12.0000000
generate(dist, 10)
#> [[1]]
#> [1] 1.09151677 0.56131180 0.09353204 0.40435461 3.40747023 2.15390359
#> [7] 6.51187518 7.41812929 3.56274747 1.33050974
#>
#> [[2]]
#> [1] 10.6266955 3.2277882 0.7163586 0.3818369 3.1841318 13.6396875
#> [7] 2.8099091 3.6857501 5.5754641 1.1196156
#>
#> [[3]]
#> [1] 0.6280509 10.3712049 18.8556150 5.7181777 3.0552349 4.9124656
#> [7] 6.0932573 6.1025130 10.3560330 2.8340610
#>
#> [[4]]
#> [1] 3.455234 7.514794 2.630157 3.904155 5.449911 8.496484 7.864584 3.785039
#> [9] 5.255440 3.672237
#>
#> [[5]]
#> [1] 6.253478 2.581500 3.207688 1.689477 4.155992 5.055745 2.891685 3.933688
#> [9] 4.462094 6.330628
#>
#> [[6]]
#> [1] 8.940611 8.131849 5.144869 10.355557 4.477071 8.163972 8.793458
#> [8] 7.730500 6.581083 6.958249
#>
#> [[7]]
#> [1] 0.37638058 0.01552343 0.42259779 0.44481944 0.23820153 0.11660630
#> [7] 0.08644417 0.40702216 0.15272538 0.61136044
#>
density(dist, 2)
#> [1] 0.183939721 0.183939721 0.091969860 0.090223522 0.059540363 0.006545958
#> [7] 0.053990967
density(dist, 2, log = TRUE)
#> [1] -1.693147 -1.693147 -2.386294 -2.405465 -2.821101 -5.028908 -2.918939
cdf(dist, 4)
#> [1] 0.8646647 0.5939942 0.3233236 0.3711631 0.4074527 0.0762173 0.9953223
quantile(dist, 0.7)
#> [1] 2.4079456 4.8784330 7.2311353 5.8903613 5.1503385 8.6608472 0.5370971