20065ee131A_1_EE131AF06HW3

20065ee131A_1_EE131AF06HW3 - Assignment 3 EE 131A Due...

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Assignment 3, EE 131A Due: Wednesday October 25, 2006 Problem 1. Consider a binary communication channel where the probabilities of inputs “0” and “1” into the system are p and 1 - p , respectively. Suppose that the transmission errors occur at random with probability ε . For i = 0 , 1 , let A i be the event “input was i ,” and B i be the event “output was i .” Thus P [ A 0 ] = p , P [ A 1 ] = 1 - p , P [ B 0 | A 0 ] = 1 - ε , P [ B 1 | A 0 ] = ε , P [ B 0 | A 1 ] = ε , and P [ B 1 | A 1 ] = 1 - ε . Find the value of ε for which the input of the channel is independent of the output of the channel. Problem 2. Suppose that for the general population, 1 in 5000 people carries the human immun- odeficiency virus (HIV). A test for the presence of HIV yields either a positive ( + ) or negative ( - ) response. Suppose the test gives the correct answer 99% of the time. What is P [ -| H ] , the conditional probability that a person tests negative given that the person does have the HIV virus? What is P [ H | +] , the conditional probability that a randomly chosen person has the HIV virus given that the person tests positive?
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