In this problem, we repeat the voice/data line detection test of Problem 8.2.6, except now we observe n calls from one line and records whether each call lasts longer than t0 minutes. The random variable K is the number of calls that last longer than t0 minutes. The decision is H0 if K ≤ k0. Otherwise, the decision is H1.
(a) Write a formula for the false alarm probability as a function of t0, k0, and n.
(b) Find the maximum likelihood decision number k0 = kML for t0 = 4.5 minutes and n = 16 calls monitored
(c) Find the maximum a posteriori probability decision number k0 = kMAP for t0 = 4.5 minutes and n = 16 calls monitored.
(d) Draw the receiver operating curves for t0 = 4.5 minutes and t0 = 3 minutes. In both cases let n = 16 calls monitored
Problem 8.2.6
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 Mn(T), the sample mean of the duration of a call. The decision is H0 if Mn (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.