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Unformatted text preview: University of Illinois Fall 2010 ECE 313: Problem Set 4: Solutions Confidence intervals, ML parameter estimation, Bernoulli processes, Poisson Distribution 1. [Binomial Random Variable] (a) The probability that the aircraft will fail on a flight is given by 4 X k =2 4 k (10- 3 ) k (1- 10- 3 ) 4- k = 6 10- 6 + 4 10- 9 + 10- 12 6 10- 6 (b) Let n be the number of flights. The aircraft can crash with a probability of p = 10- 9 in any given flight. As each flight is independent of the other, and if X represents the number of crashes in n flights, then its pmf is binomially distributed with parameters ( n,p ). Thus, the probability of having at least one crash in n flights is given by 1- ( n ) (1- p ) n = 1- (1- p ) n . Searching over values of n starting from 1, we find that n = 100 , 006 flights are needed in order for the probability of the aircraft experiencing at least one crash reaches 0 . 01%. (c) Sweeping p c from say 0 upwards in the equation below until it evaluates to 10- 9 : 4 X k =2 4 k ( p c ) k (1- p c ) 4- k we get p c = 1 . 35...
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