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 Î» Î» Exp U X ~ log 1- = ( 29 Î» Exp X X iid n ~ ,..., 1 ( 29 Î» , ~ 1 n X X Y n Î“ + + = ( 29 Î» Î± , ~ Î“ Y Î“ = a aY W Î» Î± , ~ ( 29 1 , ~ N Z Î“ = 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 âˆˆ =-- , 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 â€“ 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 Î»- = ( 29 ( 29 âˆž âˆˆ â‡’ âˆˆ , 1 , y x y e x x y Î» Î»- = â‡”- = log 1 ( 29 ( 29 y X Y e dy dx x f y f Î» Î»-- Ã— = = 1 ( 29 ( 29 âˆž âˆˆ =- , , y e y f y Y Î» Î» ( 29 Î» Î» Exp X Y ~ log 1- = Transformation of Random Variables Example ( 29 2 , ~ Ïƒ Î¼ N X X e Y = ( 29 ( 29 âˆž âˆˆ â‡’ âˆž âˆž- âˆˆ , , y x y x e y x log = â‡” = ( 29 ( 29 ( 29 y y dy dx x f y f X Y 1 2 log exp 2 1 2 2 2 Ã— -- = = Ïƒ Î¼ Ï€Ïƒ ( 29 ( 29 ( 29 âˆž âˆˆ -- = , , 2 log exp 2 1 2 2 2 y y y y f Y Ïƒ Î¼ Ï€Ïƒ ( 29 2 , ~ Ïƒ Î¼ LogNormal e Y X = multiplicative product of random quantities such as return rate Transformation of Random Variables Example ( 29 Î» Exp X ~ , 1 = Î² Î² X Y ( 29 ( 29 âˆž âˆˆ â‡’ âˆž âˆˆ , , y x Î² Î² y x x y = â‡” = 1 ( 29 ( 29 1-- Ã— = = Î² Î» Î² Î» Î² y e dy dx x f y f y X Y ( 29 ( 29 âˆž âˆˆ =-- , , 1 y e y y f y Y Î² Î» Î² Î»Î² ( 29 Î» Î² Î² , ~ 1 Weibull X Y = lifetime model of survival analysis in life science Transformation of Random Variables Example Î“ Î» , 2 3 ~ X X Y = ( 29 ( 29 âˆž âˆˆ â‡’ âˆž âˆˆ , , y x 2 y x x y = â‡” = ( 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 Ã— Î“ = =-- Î» Î» ( 29 ( 29 âˆž âˆˆ =- , , 4 2 2 2 3 y e y y f y Y Î» Ï€ Î» model particle speeds in ideal gas at equilibrium Maxwell-Boltzmann Transformation of Random Variables Example ( 29 1 , ~ N X 2 X Y = ( 29 ( 29 âˆž âˆˆ â‡’ âˆž âˆž- âˆˆ , , y x y x x y Â± = â‡” = 2 ( 29 ( 29 dy dx x f y f X Y = g â€“1 not exists Transformation of Random Variables Example ( 29 1 , ~ N X 2 X Y = ( 29 ( 29 âˆž âˆˆ â‡’ âˆž âˆž- âˆˆ , , y x ( 29 ( 29 ( 29 ( 29 y X y P y X P y Y P y F y â‰¤ â‰¤- = â‰¤ = â‰¤ = 2 ( 29 ( 29 , 1 2- Î¦ = 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 ' Ã— - Ã— = = Ï€ ( 29 , 2 1 2 1 =- y e y y f y Y Ï€ ( 29 ( 29 ( 29 , 2 1 2 1 2 1 2 1 2 1 Î“ =-- y e y y f y Y Î“ = 2 1 , 2 1 ~ 2 X Y Chi-square distribution with 1 degree of freedom 2 1 ~ Ï‡ 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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