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# prob - Exponential Distribution Definition X is said to...

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Unformatted text preview: Exponential Distribution Definition X is said to have an exponential distribution with parameter Î» ( Î» > 0) if the pdf of X is f ( x ; Î» ) = ( Î» e- Î» x x â‰¥ otherwise Remark: 1. Usually we use X âˆ¼ EXP( Î» ) to denote that the random variable X has an exponential distribution with parameter Î» . 2. In some sources, the pdf of exponential distribution is given by f ( x ; Î¸ ) = ( 1 Î¸ e- x Î¸ x â‰¥ otherwise The difference is that Î» â†’ 1 Î¸ . Liang Zhang (UofU) Applied Statistics I June 30, 2008 1 / 20 Exponential Distribution Liang Zhang (UofU) Applied Statistics I June 30, 2008 2 / 20 Exponential Distribution Proposition If X âˆ¼ EXP ( Î» ) , then E ( X ) = 1 Î» and V ( X ) = 1 Î» 2 And the cdf for X is F ( x ; Î» ) = ( 1- e- Î» x x â‰¥ x < Liang Zhang (UofU) Applied Statistics I June 30, 2008 3 / 20 Exponential Distribution Proof: E ( X ) = Z âˆž x Î» e- Î» x dx = 1 Î» Z âˆž ( Î» x ) e- Î» x d ( Î» x ) = 1 Î» Z âˆž ye- y dy y = Î» x = 1 Î» [- ye- y | âˆž + Z âˆž e- y dy ] integration by parts: u = y , v =- e- y = 1 Î» [0 + (- e- y | âˆž )] = 1 Î» Liang Zhang (UofU) Applied Statistics I June 30, 2008 4 / 20 Exponential Distribution Proof (continued): E ( X 2 ) = Z âˆž x 2 Î» e- Î» x dx = 1 Î» 2 Z âˆž ( Î» x ) 2 e- Î» x d ( Î» x ) = 1 Î» 2 Z âˆž y 2 e- y dy y = Î» x = 1 Î» 2 [- y 2 e- y | âˆž + Z âˆž 2 ye- y dy ] integration by parts = 1 Î» 2 [0 + 2(- ye- y | âˆž + Z âˆž e- y dy )] integration by parts = 1 Î» 2 2[0 + (- ye- y | âˆž )] = 2 Î» 2 Liang Zhang (UofU) Applied Statistics I June 30, 2008 5 / 20 Exponential Distribution Proof (continued): V ( X ) = E ( X 2 )- [ E ( X )] 2 = 2 Î» 2- ( 1 Î» ) 2 = 1 Î» 2 F ( x ) = Z x Î» e- Î» y dy = Z x e- Î» y d ( Î» y ) = Z x e- z dz z = Î» y =- e- z | x = 1- e- x Liang Zhang (UofU) Applied Statistics I June 30, 2008 6 / 20 Exponential Distribution Example (Problem 108) The article â€œDetermination of the MTF of Positive Photoresists Using the...
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