Beta Distribution - Section 4.7
The beta distribution is used to model continuous data with values between 0 and 1. Typically,
beta distributions are used to model proportions. The characteristics of the beta distribution are
•The parameters of the beta distribution are α > 0 and β > 0. Unlike the gamma distribution,
these parameters do not have special names. With different values of these parameters, the
shape of the probability density curve can change significantly.
•The probability density curve for a beta distribution is
f(y) = yα−1(1 −y)β−1
B(α, β)0≤y≤1
where B(α, β) = Γ(α)Γ(β)
Γ(α+β)
•Under most circumstances, the distribution function F(y) for the beta distribution has no
closed form solution. Therefore, in order to find probabilities associated with the beta distri-
bution, we must use tables or a statistical computer package.
Here are graphs of the probability density function and the distribution function for a beta
distribution with α= 2 and β= 3.
p.d.f. of Beta Distribution with alpha = 2 and beta = 3
y
f(y)
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5
Distribution of Beta Distribution with alpha = 2 and beta = 3
y
F(y)
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
•The theoretical mean of the beta distribution is
µ=E(Y) = α
α+β
•The variance of the beta distribution is
σ2=V(Y) = αβ
(α+β)2(α+β+ 1)
Working with the beta distribution in R.
To find the probability P(Y≤y) the command in R is
pbeta(y,alpha,beta)
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