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 ( > 0) if the pdf of X is f ( x ; ) = ( e- x x otherwise Remark: 1. Usually we use X EXP( ) to denote that the random variable X has an exponential distribution with parameter . 2. In some sources, the pdf of exponential distribution is given by f ( x ; ) = ( 1 e- x x otherwise The difference is that 1 . 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 EXP ( ) , then E ( X ) = 1 and V ( X ) = 1 2 And the cdf for X is F ( x ; ) = ( 1- e- x x x < Liang Zhang (UofU) Applied Statistics I June 30, 2008 3 / 20 Exponential Distribution Proof: E ( X ) = Z x e- x dx = 1 Z ( x ) e- x d ( x ) = 1 Z ye- y dy y = x = 1 [- ye- y | + Z e- y dy ] integration by parts: u = y , v =- e- y = 1 [0 + (- e- y | )] = 1 Liang Zhang (UofU) Applied Statistics I June 30, 2008 4 / 20 Exponential Distribution Proof (continued): E ( X 2 ) = Z x 2 e- x dx = 1 2 Z ( x ) 2 e- x d ( x ) = 1 2 Z y 2 e- y dy y = x = 1 2 [- y 2 e- y | + Z 2 ye- y dy ] integration by parts = 1 2 [0 + 2(- ye- y | + Z e- y dy )] integration by parts = 1 2 2[0 + (- ye- y | )] = 2 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 2- ( 1 ) 2 = 1 2 F ( x ) = Z x e- y dy = Z x e- y d ( y ) = Z x e- z dz z = 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 Determination of the MTF of Positive Photoresists Using the...
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This note was uploaded on 01/06/2011 for the course CHE 6270 taught by Professor Dr.dmitry during the Spring '10 term at University of Florida.

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

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