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Section_4

# Section_4 - CEE 304 Section 4 Problems 1 Transforming...

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CEE 304 – Section 4 Problems (9/20/2006) 1. Transforming Variables Example : ( ) x X f x e = for x > 0, and 0 otherwise Develop the density function for Y = X ½ Y = g(X) = X ½ => X = Y 2 , y dy dx 2 = One-to-one transformation: we have a one-to-one relationship between X and Y, so we can use the following formula to find the pdf of Y given the pdf of X: ( ) ( ) ( ) [ ] ( ) [ ] dy dx y x f dy dx dx y x dF dy y dF y f X X Y Y = = = 2 ( ) (2 ) y Y f y e y = for y > 0, and 0 otherwise 2. Memoryless Exponential Process : ( ) x X f x e λ λ = for x 0 ( ) 1 x X F x e λ = for x > 0 By definition: ( ) [ ] X F x P X x = [ ] 1 ( ) x X P X x F x e λ = = Additional lifetime (i.e. how much long it will last) = A t t a X X e e t F t a F t X P t X t a X P t X t a X P λ λ + = + = + = + ) ( ) ( 1 ) ( 1 ] [ ] [ ] | [ a e λ = By definition: ( ) ( ) t X e t F t X P λ = = 1 , for t 0 Therefore, the distribution of the additional lifetime A is exactly the same as the original distribution of waiting time T. So, E [ A ] = 1/ λ . 1

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