The ping time, in milliseconds of a ne\v transmission

Chapter 11, Problem 11.15

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The ping time, in milliseconds of a ne\v transmission system, described in 11.1.4 is the exponential (60) random variable N . T he ping time of an old syste1n is an exponential random variable 0 with expected value ,o > 60 ms. The null hypothesis of a binary hypothesis test is Ho: The transmission syste1n is the new system. The alternative hypothesis is H i : The transmission system is the old system. The probability of a new system is P[N] = 0.8. T he probability of an old system is P [O] = 0.2. A binary hypothesis test measures T milliseconds, the result of one ping test. T he decision is Ho if T < to ms. Other,vise, the decision is H 1 . (a) Write a formula for t he false alarm probability as a function of to and ,o . (b) Write a formula for the miss probability as a function of t0 and ,o . ( c) Calculate the maximum likelihood decision time t0 = t j\1 L for ,o = 120 ms and ,o = 200 ms. (d) Do you think that trvrAP, the maximum a posteriori decision time, is greater than or less than t rv1L? Explain your answer. ( e) Calculate the maximum a posteriori probability decision time to= tMAP for ,o = 120 ms and ,o = 200 ms. (f) Dra'v the receiver operating curves for ,o = 120 ms and o = 200 ms.