The duration of a voice telephone call is an exponential

Question

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] = µ_D> 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 µ_D.

(b) Write a formula for the miss probability as a function of t₀ and µ_D.

(c) Calculate the maximum likelihood decision time t₀ = t_{M L} for µ_D = 6 minutes and µ_D = 10 minutes.

(d) Do you think that t_MAP, the maximum a posteriori decision time, is greater than or less than t_ML? Explain your answer.

(e) Calculate the maximum a posteriori probability decision time t₀ = t_MAP for µ_D = 6 minutes and µ_D = 10 minutes.

(f) Draw the receiver operating curves for µ_D = 6 minutes and µ_D = 10 minutes.