Section_4 - CEE 304 Section 4 Problems (9/20/2006) 1....

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CEE 304 – Section 4 Problems (9/20/2006) 1. Transforming Variables Example : () x X f xe = 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 ye y =⋅ for y > 0, and 0 otherwise 2. Memoryless Exponential Process : x X f λ = for x 0 () 1 x X F =− for x > 0 By definition: ( ) [ ] X Fx PX x =≤ 1 ( ) x X PX x F x e ≥= = 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 + = + = + = + ) ( ) ( 1 ) ( 1 ] [ ] [ ] | [ a e = By definition: ( ) t X e t F t X P = = 1, f o r 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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3. An engineering student got a summer job setting off fireworks on the 4 th of July and other summer events. For a particular display, the bomb is shot from the tube so that after t seconds it is at height: H = (100 m/sec) t – 0.5 (10 m/sec 2 ) t 2 = 500 – 5(10 – t) 2 meters Unfortunately the timers on the bombs are not accurate. The student estimates that the timer
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This note was uploaded on 02/02/2008 for the course CEE 3040 taught by Professor Stedinger during the Fall '08 term at Cornell University (Engineering School).

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Section_4 - CEE 304 Section 4 Problems (9/20/2006) 1....

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