Spring2010ExamOne

# Spring2010ExamOne - Rensselaer Electrical Computer and...

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Rensselaer Electrical, Computer, and Systems Engineering Department ECSE-4500 PEA Exam 1 Solutions March 19, 2010 Closed book - One sheet of notes allowed. Please show work for full credit. Only hand in your answer booklet . Good luck! 1. (30%) Often we are faced with determining the more likely of two alternatives. In such a case we are given two probability measures for a single sample space and f eld of events, i.e. ( , F ,P 1 ) and ( , F 2 ) , and we are asked to determine the probability of an observed event E in both cases. The more likely alternative is said to be the one which gives the higher probability of E . Consider that two coins are in a box, one is ’fair’ with P 1 [ { H } ]=0 . 5 and one is ’biased’ with P 2 [ { H } ]= p . Without looking, we draw one coin from the box and then F ip this single coin 10 times. We only consider the repeated coin- F ipsasourexper imentandsothesamp lespace = { all ten-character strings of H and T } . We observe the event E = { a total of four Hs and six Ts } . (a) (20%) What are the two probabilities of the observed event E, i.e. P 1 [ E ] and P 2 [ E ] ? Ans: This is Bernoulli trials. Thus P 1 [ E ¡ 10 4 ¢¡ 1 2 ¢ 10 and P 2 [ E ¡ 10 4 ¢ p 4 (1 p ) 6 . (b) (10%) Determine the likelihood ratio L , P 1 [ E ] /P 2 [ E ] as a function of p. (When L> 1 we say that the fair coin is more likely. This test is called a

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Spring2010ExamOne - Rensselaer Electrical Computer and...

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