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The duration of a voice telephone call is an exponential random variable V with expected value E[V] = 3 minutes. The duration of a data call is an exponential random variable D with expected value E[D] = µD> 3 minutes. The null hypothesis of a binary hypothesis test is H0: a call is a voice call. The alternative hypothesis is H1: a call is a data call. The probability of a voice call is P[V ] = 0.8. The probability of a data call is P[D] = 0.2. A binary hypothesis test measures T minutes, the duration of a call. The decision is H0if T ≤ t0 minutes. Otherwise, the decision is H1.
(a) Write a formula for the false alarm probability as a function of t0 and µD.
(b) Write a formula for the miss probability as a function of t0 and µD.
(c) Calculate the maximum likelihood decision time t0 = tM L for µD = 6 minutes and µD = 10 minutes.
(d) Do you think that tMAP, the maximum a posteriori decision time, is greater than or less than tML? Explain your answer.
(e) Calculate the maximum a posteriori probability decision time t0 = tMAP for µD = 6 minutes and µD = 10 minutes.
(f) Draw the receiver operating curves for µD = 6 minutes and µD = 10 minutes.
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