Problem # 2 We have a binary channel corrupted by noise and the following inform

Question

Problem # 2 We have a binary channel corrupted by noise and the following information is provided. P(O)-0.6 and P(1| 1)-P(olO-0.8. If we have 3 cascaded channels, what is the probability of receiving a '1' at the output? lf at t he output of first channel '1' was received, what is the probability that it was transmitted ? What is the probability that it was actually transmitted as a 1? What is the Probability of Errori? Problem # 3 In a quaternary communication system, a '4' is transmitted 4 times as frequently as a '1', a '3' is transmitted 3 times as frequently as a '1', and a '2' is transmitted twice as frequently as a 1'. The values are received as they are 70% of the time and the remaining time, they may be received as any of the other three. If a '4' is observed, what is the probability that it was transmitted as a '2'? Problem # 4 Data collected from a driverless car experiment is given (See next page). There are 40 measurements when there is a target present and 25 measurements when no target is present. Because of errors in measurement, a threshold was set at 2.4 (where the two histograms meet). What is PPV?

Explanation / Answer

SOlving question 1

P(0) = 0.6, P(1)=0.4

Let P( receiving 0 if 0) be a and P( receiving 1 if 0 ) be b

P( 1) at output = P(001)+P(011)+P(101)+p(111) = 0.6*0.8*0.2+0.6*0.2*0.8+ 0.4*0.2*0.2 +0.4*0.8*0.8 = 0.464

P( 1 at first channel and 0 at third channel) = P(110)+P(100)/P(1) = (0.4*0.8*0.2+0.4*0.2*0.8) / 0.4 = 0.32

P( 1 at first channel and 1 at output) = P( 111)+P(101) /P(1) = (0.4 * 0.8*0.8 + 0.4*0.2*0.2)/0.4 = 0.68

P( Error) = 0.32

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