University of California, Los AngelesDepartment of StatisticsStatistics 100A Instructor: Nicolas ChristouBeta distributionThe beta density function is defined over the interval 0 ≤ x ≤ 1 and it can be used tomodel proportions (e.g. the proportion of time a machine is under repair, the proportion ofa certain impurity in a chemical product, etc.). The probability density function of the betadistribution is given by:f(x) =xα−1(1 − x)β−1B(α, β), α > 0, β > 0, 0 ≤ x ≤ 1.where,B(α, β) =Z10xα−1(1 − x)β−1dx.The shape of the distribution depends on the values of the parameters α and β. When α = βthe distribution is symmetric about12as shown in the figure below:0.0 0.2 0.4 0.6 0.8 1.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5xf(x)Beta distribution densities with parameters αα == ββαα == 10αα == 3αα == 1αα ==141When α > β the distribution is skewed to the left and when α < β it skewed to the right(see next figure).0.0 0.2 0.4 0.6 0.8 1.00.0 0.5 1.0 1.5 2.0 2.5xf(x)Beta distribution densities with parameters αα >> ββ and αα << ββαα == 6 , ββ == 4.5αα == 1.5 , ββ == 3Even though x was defined in the interval 0 ≤ x ≤ 1, its use can be extended to randomvariables defined over some finite interval, c ≤ x ≤ d. In this case we can simply rescale thevariable using y =x−cd−c, and y will be between 0 and 1.It can be shown thatB(α, β) =Γ(α)Γ(β)Γ(α + β).Using this relation between the beta and gamma functions we can find the mean and varianceof the beta distribution:E(X) =αα + βandvar(X) =αβ(α + β)2(α + β +
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