A common method for detecting a signal in the prescence of noise is to establish a threshold level and compare the value of any observation witht this threshold. If the threshold is exceeded, it is decided that signal is present. Sometimes noise alone will exceed the threshold and this is known as a "false alarm." Usually it is desired to keep the probability of a false alarm very small. At the same time, we would like any observation that does contain a signal plus the noise to exceed the threshold with a large probability. This is the probability of detection should be as close to 1 as possible. Assume that the noise is a normal (gaussian) random variable, N, with E[N] = u_N = 0 and variance = 1V^2 and the threshold is equal to 2 volts
a) find the probability of a false alarm
b) if an observation has a calue of 4 volts in the prescence of noise, what is the probability of detection (that is, there is a valid signal present that exceeds the threshold voltage)?