Some telephone lines are used only for voice calls. Others are connected to modems and used only for data calls. 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 = 6 minutes. The null hypothesis of a binary hypothesis test is H₀ : a line is used for voice calls. The alternative hypothesis is H₁: a line is a data line. The probability of a voice line is P[V] = 0.8. The probability of a data line is P[D] = 0.2.

A binary hypothesis test observes n calls from one telephone line and calculates M_n(T), the sample mean of the duration of a call. The decision is H0 if M_n (T) ≤ t₀ minutes. Otherwise, the decision is H₁.

(a) Use the central limit theorem to write a formula for the false alarm probability as a function of t₀and n.

(b) Use the central limit theorem to write a formula for the miss probability as a function of t₀ and n.

(c) Calculate the maximum likelihood decision time, t₀ = tML, for n = 9 calls monitored.

(d) Calculate the maximum a posteriori probability decision time, t₀ = tMAP for n = 9 calls monitored.

(e) Draw the receiver operating curves for n = 9 calls monitored and n = 16 calls monitored.