section 4doc - CEE 3040 Fall 2008 CEE 304 Section 4...

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CEE 3040 Fall 2008 CEE 304 – Section 4 Problems 1. Memoryless Exponential Process : ( ) x X f x e λ - = for x 0 ( ) 1 x X F x e - = - for x > 0 By definition: ( ) [ ] X F x P X x = [ ] 1 ( ) x X P X x F x e - = - = If we already waited t minutes, how much longer do we expect to wait? [i.e. What is the mean of the additional waiting time A ?] t t a X X A e e t F t a F t X P t X t a X P t X t a X P a F - + - = - + - = + = + = ) ( ) ( 1 ) ( 1 ] [ ] [ ] | [ ) ( a A e a F - = ) ( , for a > t Therefore, the distribution of the additional waiting time A is the same as the original waiting time T, it is just shifted over by t . So, E [ A ] = E [X | X t ] = t + 1/ λ . Doesn’t matter how long you have waited, the expected additional waiting time has mean 1/ λ . 2. 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 will yield a time T to detonation after launch that is normally distributed with mean 10 seconds and standard deviation 1.2 seconds. T ~ N( μ = 10 sec, σ 2 = (1.2 sec) 2 ) a.) What is the probability that the bomb explodes more than 495 meters above the ground? ] 495 ) 10 ( 5 500 [ ] 495 [ 2 - - = t P H P = 0.5934 b.) Above what height can you be 99% confident the bomb will explode? 99 . 0 ] [ = h H P 99 . 0 ] ) 10 ( 5 500 [ 2 = - - h t P Ans . h = 452.26 m
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CEE 3040 Fall 2008 c.) What is the pdf for the height at which the bomb explodes? [We can easily derive the cdf for the height of explosion F H (h), but deriving the pdf f H (h) is made difficult by the need to evaluate d Φ (h)/dh. When originally asked on an exam, this
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This note was uploaded on 10/13/2010 for the course CEE 3040 at Cornell University (Engineering School).

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section 4doc - CEE 3040 Fall 2008 CEE 304 Section 4...

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