The ping time, in milliseconds, of a new transmission system, described in Problem 11.1.4 is the exponential (60) random variable N. The ping time of an old system is the exponential (120) random variable O. The null hypothesis of a binary hypothesis test is H0: The transmission system is the new system. The alternative hypothesis is H1. The transmission system is the old system. The probability of a new system is P[N] = 0.8. The probability of an old system P[0] = 0.2. A binary hypothesis test performs k ping tests and calculates Mn(T), the sample mean of the ping time. The decision is H0 if Mn(T) ≤ t0 ms. Otherwise, the decision is H1.
(a) Use the central limit theorem to write a formula for the false alarm probability as a function of t0 and k.
(b) Use the central limit theorem to write a formula for the miss probability as a function of t0 and k.
(c) Calculate the maximum likelihood decision time, t0 = tML, for k = 9 ping tests.
(d) Calculate the maximum a posteriori probability decision time, to = tMAP for k = 9 ping tests.
(e) Draw the receiver operating curves for k = 9 ping tests and k = 16 ping tests.