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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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- Fall '07
- Statistics, Probability theory, probability density function, α, Nicolas Christou, Beta distribution, Beta distribution densities