The Gamma distribution

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 = 1/scale, scale = 1/rate)

Arguments

shape, scale: shape and scale parameters. Must be positive, scale strictly.
rate: an alternative way to specify the scale.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_gamma.html

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):

$$ E(e^{tX}) = \Big(\frac{\beta}{ \beta - t}\Big)^{\alpha}, \thinspace t < \beta $$

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
#> <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] 2.5425472 2.4326614 0.7754665 3.5466399 0.3745336 1.0187754 1.3440671
#>  [8] 0.2341321 2.5564569 1.6291680
#> 
#> [[2]]
#>  [1]  6.136013  1.793628  2.333732  2.907564  1.689938  7.467933  6.328497
#>  [8] 10.656851 11.386213  1.021115
#> 
#> [[3]]
#>  [1] 4.6538120 2.8251657 0.7241433 4.2995857 2.2476634 0.6490697 7.8635906
#>  [8] 7.1416508 4.2418193 1.3142215
#> 
#> [[4]]
#>  [1] 4.986279 3.379855 6.585655 1.904162 6.649301 5.613127 5.335366 2.393918
#>  [9] 5.167640 4.119382
#> 
#> [[5]]
#>  [1] 4.208062 2.194900 3.232550 5.157432 5.861315 4.429140 2.823739 5.793119
#>  [9] 3.821173 3.960008
#> 
#> [[6]]
#>  [1] 6.801574 6.593407 5.387267 8.144641 7.304945 7.170135 4.778968 4.907044
#>  [9] 8.879577 7.416681
#> 
#> [[7]]
#>  [1] 0.425867092 0.471833324 0.576314651 1.652474344 0.004832496 0.322129095
#>  [7] 0.913436191 0.001937892 0.031468788 0.743272721
#> 

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

Arguments

Details

See also

Examples