The Beta distribution

The Beta distribution is a continuous probability distribution defined on the interval [0, 1], commonly used to model probabilities and proportions.

dist_beta(shape1, shape2)

Arguments

shape1, shape2: The non-negative shape parameters of the Beta distribution.

Details

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

In the following, let $X$ be a Beta random variable with parameters shape1 = $\alpha$ and shape2 = $\beta$.

Support: $x \in [0, 1]$

Mean: $\frac{\alpha}{\alpha + \beta}$

Variance: $\frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}$

Probability density function (p.d.f):

$$ f(x) = \frac{x^{\alpha - 1}(1-x)^{\beta - 1}}{B(\alpha, \beta)} = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha - 1}(1-x)^{\beta - 1} $$

where $B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$ is the Beta function.

Cumulative distribution function (c.d.f):

$$ F(x) = I_x(alpha, beta) = \frac{B(x; \alpha, \beta)}{B(\alpha, \beta)} $$

where $I_x(\alpha, \beta)$ is the regularized incomplete beta function and $B(x; \alpha, \beta) = \int_0^x t^{\alpha-1}(1-t)^{\beta-1} dt$.

Moment generating function (m.g.f):

The moment generating function does not have a simple closed form, but the moments can be calculated as:

$$ E(X^k) = \prod_{r=0}^{k-1} \frac{\alpha + r}{\alpha + \beta + r} $$

Examples

dist <- dist_beta(shape1 = c(0.5, 5, 1, 2, 2), shape2 = c(0.5, 1, 3, 2, 5))

dist
#> <distribution[5]>
#> [1] Beta(0.5, 0.5) Beta(5, 1)     Beta(1, 3)     Beta(2, 2)     Beta(2, 5)    
mean(dist)
#> [1] 0.5000000 0.8333333 0.2500000 0.5000000 0.2857143
variance(dist)
#> [1] 0.12500000 0.01984127 0.03750000 0.05000000 0.02551020
skewness(dist)
#> [1]  0.000000 -5.916080  2.581989  0.000000  5.962848
kurtosis(dist)
#> [1] -1.5000000  1.2000000  0.0952381 -0.8571429 -0.1200000

generate(dist, 10)
#> [[1]]
#>  [1] 0.51272391 0.93824073 0.99515628 0.04294426 0.36260647 0.31435554
#>  [7] 0.00302847 0.41149750 0.60539748 0.99916715
#> 
#> [[2]]
#>  [1] 0.7617427 0.7515718 0.9677268 0.8482862 0.6388981 0.8219619 0.7697629
#>  [8] 0.9109772 0.9994124 0.8433937
#> 
#> [[3]]
#>  [1] 0.115627217 0.216302708 0.074413319 0.329727704 0.316671953 0.244305850
#>  [7] 0.241259576 0.120459703 0.001220667 0.373141784
#> 
#> [[4]]
#>  [1] 0.5600205 0.7817342 0.3174363 0.7904692 0.4308882 0.1958695 0.2407125
#>  [8] 0.7796609 0.5371046 0.6358395
#> 
#> [[5]]
#>  [1] 0.05643909 0.34659362 0.62582410 0.33593872 0.40339891 0.42415031
#>  [7] 0.26402567 0.32176762 0.26781893 0.61462242
#> 

density(dist, 2)
#> [1] 0 0 0 0 0
density(dist, 2, log = TRUE)
#> [1] -Inf -Inf -Inf -Inf -Inf

cdf(dist, 4)
#> [1] 1 1 1 1 1

quantile(dist, 0.7)
#> [1] 0.7938926 0.9311499 0.3305670 0.6367425 0.3603577

Arguments

Details

See also

Examples