University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou
Beta distribution
The beta density function is defined over the interval 0 ≤ x ≤ 1 and it can be used to model proportions (e.g. the proportion of time a machine is under repair, the proportion of a certain impurity in a chemical product, etc.). The probability density function of the beta distribution is given by:
xα−1(1 − x)β−1
f(x)= B(α,β) , α>0, β>0, 0≤x≤1.
Statistics 100B
where,
B(α, β) =
The shape of the distribution depends on the values of the parameters α and β. When α = β the distribution is symmetric about 1 as shown in the figure below:
2
Beta distribution densities with parameters α = β
1 α−1 β−1
x (1 − x) dx.
0
1
α =
4
α=10
α=3
α=1
0.0 0.2 0.4
0.6
0.8 1.0
x
1
0.0 0.5 1.0 1.5
2.0 2.5 3.0 3.5
f(x)
When α > β the distribution is skewed to the left and when α < β it skewed to the right (see next figure). Beta distribution densities with parameters α > β and α < β α = 1.5 , β = 3 α = 6 , β = 4.5 of the beta distribution: E(X)= α 0.0 0.2 0.4 0.6 0.8 1.0 x Even though x was defined in the interval 0 ≤ x ≤ 1, its use can be extended to random variables defined over some finite interval, c ≤ x ≤ d. In this case we can simply rescale the variable using y = x−c , and y will be between 0 and 1. d−c It can be shown that B(α,β) = Γ(α)Γ(β). Γ(α+β) Using this relation between the beta and gamma functions we can find the mean and variance and α+β var(X) = αβ . (α + β)2(α + β + 1) 2 0.0 0.5 1.0 1.5 2.0 2.5 f(x)