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lect18 - BETA DISTRIBUTION CHANGE OF VARIABLE FORMULA FOR...

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BETA DISTRIBUTION CHANGE OF VARIABLE FORMULA FOR BIVARIATE DENSITIES Outline BETA DISTRIBUTION CHANGE OF VARIABLE FORMULA FOR BIVARIATE DENSITIES Bivariate to Univariate Transformation Bivariate to Bivariate Transformation 1 / 12 Xinghua Zheng Lect 18: Beta Distribution; Change of Variables in 2D
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BETA DISTRIBUTION CHANGE OF VARIABLE FORMULA FOR BIVARIATE DENSITIES Minimum and Maximum Suppose U 1 , U 2 are i.i.d. standard uniforms. Let X := min ( U 1 , U 2 ) , Y := max ( U 1 , U 2 ) Their joint density: ( X , Y ) takes values in f X , Y ( x , y ) dxdy = P ( x X x + dx , y Y y + dy ) = , so f X , Y ( x , y ) = , . Marginal densities: 1. f X ( x ) = , 2. f Y ( y ) = , 2 / 12 Xinghua Zheng Lect 18: Beta Distribution; Change of Variables in 2D
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BETA DISTRIBUTION CHANGE OF VARIABLE FORMULA FOR BIVARIATE DENSITIES The Beta Distribution Suppose 0 < α, β < . A random variable X with density f X ( x ) = Γ( α + β ) Γ( α )Γ( β ) x α - 1 ( 1 - x ) β - 1 , 0 < x < 1 , is said to have a Beta distribution with parameters α and β , written as X Beta ( α, β ) . Examples: 1. If U has a standard uniform distribution, then U Beta ( , ) . 2. Suppose U 1 , U 2 are i.i.d. standard uniform random variables, and let X := min ( U 1 , U 2 ) , Y := max ( U 1 , U 2 ) . Then X , Y . 3 / 12 Xinghua Zheng Lect 18: Beta Distribution; Change of Variables in 2D
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BETA DISTRIBUTION CHANGE OF VARIABLE FORMULA FOR BIVARIATE DENSITIES Mean And Variance Of Beta Distributions Suppose X Beta ( α, β ) . Then E ( X ) = Z 1 0 xf ( x ) dx = Γ( α + β ) Γ( α )Γ( β ) Z 1 0 x · x α - 1 ( 1 - x ) β - 1 dx = = α α + β . Similarly, one can compute Var ( X ) = αβ ( α + β ) 2 ( α + β + 1 ) . Examples: 1. If U has a standard uniform distribution, then E ( U ) = .
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