hw4+fall+2011+solutions

hw4+fall+2011+solutions - ECI114 HW 4 Solutions Due Nov...

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ECI114 HW 4 Solutions Due Nov 7 th , 5pm 1. (2+4+2=8 points) Suppose that earthquakes occur randomly in the western region of the US, with an average of 2 per week. For each part, clearly define the random variable of interest. a) Find the probability that at least 3 earthquakes occur during the next 2 weeks. b) Find the probability distribution of the time, starting from now, until the next earth- quake. Give the expected value and variance of this time. c) Find the probability that no earthquakes occur for at least 15 days after an earthquake. Solution: a) From the given information, we know it is a Poisson process, so we have: = = - elsewhere x x T e x p x T 0 ,... 2 , 1 , 0 ! ) 2 ( ) ( 2 , where λ =2 quakes per week. The problem asks for Pr[X 3 for T = 2 weeks], so: Pr[X 3] = 1 – Pr[X<3] = 1- p(0) - p(1) - p(2) = 2 4 4 4 4 1 4 2 e e e - - - - - - = 4 1 13 e - - = 0.762 b) Let Y denote the time in weeks from now till the next earthquake. It follows the exponential distribution with λ = 2, so we get: 2 2 0 ( ) 0, . y e y f y elsewhere - = The expected value and variance of this time are: E(Y) = 1 = 1/2 = 0.5 week between quakes Var(Y) = 2 1 = ¼ = 0.25 weeks 2 c) To be consistent with the units of λ , we need to express the time in weeks, i.e. 15 days = 15/7 weeks. Then we want Pr[Y 15 7 ] = 1- F( 15 7 ) = 1- ( 15 2* 7 1 e - - ) = 0.0138. Alternatively, this can be done with the Poisson distribution: Pr[no quakes for at least 15 days] = Pr(X=0 for T=15/7) = e -2(15/7) [2(15/7)] 0 / 0! = e -2(15/7) = 0.0138. 1
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2. (4 + 4 + 4 = 12 pts) Consider the following CDF: < + < < = y y y y y y y F 9 1 9 4 64 . 0 04 . 0 4 0 2 . 0 0 0 ) ( a) Completely specify, and plot, the pdf. b) Find Pr [Y > 3] and Pr[3 < Y < 6]. c)
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This note was uploaded on 01/19/2012 for the course ECI 114 taught by Professor Mocktarian during the Fall '08 term at UC Davis.

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hw4+fall+2011+solutions - ECI114 HW 4 Solutions Due Nov...

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