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, 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 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):

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

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] 0.8345386 1.1237203 1.9912855 1.7387405 1.8263669 2.4061654 4.2685197
#>  [8] 0.5970121 2.1354175 2.3394327
#> 
#> [[2]]
#>  [1] 5.5980254 7.7570340 1.6022044 9.7730406 3.9013704 1.1466872 0.5395411
#>  [8] 4.1854739 4.8585414 2.2834344
#> 
#> [[3]]
#>  [1]  3.994849 10.122623  4.319014  5.963013  5.645592  1.918275  5.551286
#>  [8]  5.367362  2.799240  9.273086
#> 
#> [[4]]
#>  [1] 2.255599 6.973430 3.596424 4.088997 3.596269 4.670547 2.225786 5.315586
#>  [9] 7.903940 2.848568
#> 
#> [[5]]
#>  [1] 5.089904 5.089065 3.342935 5.122223 3.493015 3.186011 4.840678 5.673195
#>  [9] 6.182160 6.144351
#> 
#> [[6]]
#>  [1]  4.461210  8.718259  5.460712 10.774594  6.341229  6.779975  4.469242
#>  [8]  6.438585  7.619538  8.741852
#> 
#> [[7]]
#>  [1] 4.174499e-01 1.846431e+00 4.063527e-02 5.083619e-01 6.189523e-02
#>  [6] 4.118881e-01 1.912114e-01 9.462338e-02 2.659347e-03 2.335614e-05
#> 

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