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09 Beta distribution

# 09 Beta distribution - α< β it skewed to the right(see...

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University of California, Los Angeles Department of Statistics Statistics 100A 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: f ( x ) = x α - 1 (1 - x ) β - 1 B ( α,β ) , α > 0 , β > 0 , 0 x 1 . where, B ( α,β ) = Z 1 0 x α - 1 (1 - x ) β - 1 dx. The shape of the distribution depends on the values of the parameters α and β . When α = β the distribution is symmetric about 1 2 as shown in the ﬁgure below: 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x f(x) Beta distribution densities with parameters α = β α = 10 α = 3 α = 1 α = 1 4 1

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When α > β the distribution is skewed to the left and when
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Unformatted text preview: α < β it skewed to the right (see next ﬁgure). 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 x f(x) Beta distribution densities with parameters α β and α < β α = 6 , β = 4.5 α = 1.5 , β = 3 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 d-c , and y will be between 0 and 1. 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 of the beta distribution: E ( X ) = α α + β and var ( X ) = αβ ( α + β ) 2 ( α + β + 1) . 2...
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09 Beta distribution - α< β it skewed to the right(see...

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