stat1801_5

stat1801_5 - Transformation of Random Variables Functions...

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Unformatted text preview: Transformation of Random Variables Functions of random variables ( 29 1 , ~ U U ( 29 λ λ Exp U X ~ log 1- = ( 29 λ Exp X X iid n ~ ,..., 1 ( 29 λ , ~ 1 n X X Y n Γ + + = ( 29 λ α , ~ Γ Y Γ = a aY W λ α , ~ ( 29 1 , ~ N Z Γ = 2 1 , 2 1 ~ 2 Z W Transformation of Random Variables Theorem ( 29 ( 29 ( 29 ( 29 ( 29 S g y y g dy d y g f y f X Y ∈ =-- , 1 1 Let X be a continuous random variable distributed on a space S with pdf f X ( x ). Let Y = g ( X ) where g is a 1-1 function such that g – 1 exists. Then the pdf of Y can be obtained by Transformation of Random Variables x y ( 29 x f X ( 29 y f Y ( 29 X g Y = ( 29 dx x f X ( 29 dy y f Y = ( 29 ( 29 dy dx x f y f X Y = Transformation of Random Variables Example ( 29 1 , ~ U X X Y log 1 λ- = ( 29 ( 29 ∞ ∈ ⇒ ∈ , 1 , y x y e x x y λ λ- = ⇔- = log 1 ( 29 ( 29 y X Y e dy dx x f y f λ λ-- × = = 1 ( 29 ( 29 ∞ ∈ =- , , y e y f y Y λ λ ( 29 λ λ Exp X Y ~ log 1- = Transformation of Random Variables Example ( 29 2 , ~ σ μ N X X e Y = ( 29 ( 29 ∞ ∈ ⇒ ∞ ∞- ∈ , , y x y x e y x log = ⇔ = ( 29 ( 29 ( 29 y y dy dx x f y f X Y 1 2 log exp 2 1 2 2 2 × -- = = σ μ πσ ( 29 ( 29 ( 29 ∞ ∈ -- = , , 2 log exp 2 1 2 2 2 y y y y f Y σ μ πσ ( 29 2 , ~ σ μ LogNormal e Y X = multiplicative product of random quantities such as return rate Transformation of Random Variables Example ( 29 λ Exp X ~ , 1 = β β X Y ( 29 ( 29 ∞ ∈ ⇒ ∞ ∈ , , y x β β y x x y = ⇔ = 1 ( 29 ( 29 1-- × = = β λ β λ β y e dy dx x f y f y X Y ( 29 ( 29 ∞ ∈ =-- , , 1 y e y y f y Y β λ β λβ ( 29 λ β β , ~ 1 Weibull X Y = lifetime model of survival analysis in life science Transformation of Random Variables Example Γ λ , 2 3 ~ X X Y = ( 29 ( 29 ∞ ∈ ⇒ ∞ ∈ , , y x 2 y x x y = ⇔ = ( 29 ( 29 ( 29 ( 29 y e y dy dx x f y f y X Y 2 2 3 2 1 2 3 2 2 3 × Γ = =-- λ λ ( 29 ( 29 ∞ ∈ =- , , 4 2 2 2 3 y e y y f y Y λ π λ model particle speeds in ideal gas at equilibrium Maxwell-Boltzmann Transformation of Random Variables Example ( 29 1 , ~ N X 2 X Y = ( 29 ( 29 ∞ ∈ ⇒ ∞ ∞- ∈ , , y x y x x y ± = ⇔ = 2 ( 29 ( 29 dy dx x f y f X Y = g –1 not exists Transformation of Random Variables Example ( 29 1 , ~ N X 2 X Y = ( 29 ( 29 ∞ ∈ ⇒ ∞ ∞- ∈ , , y x ( 29 ( 29 ( 29 ( 29 y X y P y X P y Y P y F y ≤ ≤- = ≤ = ≤ = 2 ( 29 ( 29 , 1 2- Φ = y y y F Y ( 29 ( 29 ( 29 y y y F y f Y Y 2 1 2 1 exp 2 1 2 2 ' × - × = = π ( 29 , 2 1 2 1 =- y e y y f y Y π ( 29 ( 29 ( 29 , 2 1 2 1 2 1 2 1 2 1 Γ =-- y e y y f y Y Γ = 2 1 , 2 1 ~ 2 X Y Chi-square distribution with 1 degree of freedom 2 1 ~ χ Y Transformation of Random Variables...
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This note was uploaded on 02/01/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

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stat1801_5 - Transformation of Random Variables Functions...

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