For the random processes of Examples 10.3, 10.4, 10.5, and 10.6, identify whether the process is discrete-time or continuous-time, discrete-value or continuous-value.

Examples 10.3

Starting on January 1, we measure the noontime temperature (in degrees Celsius) at Newark Airport every day for one year. This experiment generates a sequence, C(1),C(2), . . . ,C(365), of temperature measurements. With respect to the two kinds of averages of stochastic processes, people make frequent reference to both ensemble averages such as "the average noontime temperature for February 19," and time averages, such as the "average noontime temperature for 1923.

Examples 10.4

Consider an experiment in which we record M(t), the number of active calls at a telephone switch at time t, at each second over an interval of 15 minutes. One trial of the experiment might yield the sample function m(t,s) shown in Figure 10.2. Each time we perform the experiment, we would observe some other function m(t,s). The exact m(t,s) that we do observe will depend on many random variables including the number of calls at the start of the observation period, the arrival times of the new calls, and the duration of each call. An ensemble average is the average number of calls in progress at t = 403 seconds. A time average is the average number of calls in progress during a specific 15-minute interval.

The fundamental difference between Examples 10.3 and 10.4 and experiments from earlier chapters is that the randomness of the experiment depends explicitly on time. Moreover, the conclusions that we draw from our observations will depend on time. For example, in the Newark temperature measurements, we would expect the temperatures C(1), . . . ,C(30) during the month of January to be low in comparison to the temperatures C(181), . . . ,C(210) in the middle of summer. In this case, the randomness we observe will depend on the absolute time of our observation. We might also expect that for a day t that is within a few days of t, the temperatures C(t) and C(t ) are likely to be similar. In this case, we see that the randomness we observe may depend on the time difference between observations. We will see that characterizing the effects of the absolute time of an observation and the relative time between observations will be a significant step toward understanding stochastic processes.

Figure 10.2

Examples 10.5

Suppose that at time instants T = 0, 1, 2,..., we roll a die and record the outcome N_T where 1 ≤ N_T ≤ 6. We then define the random process X(t) such that for T ≤ t _T. In this case, the experiment consists of an infinite sequence of rolls and a sample function is just the waveform corresponding to the particular sequence of rolls. This mapping is depicted on the right.

Examples 10.6

In a quaternaryphase shift keying (QPSK) communications system, one of four equally probable symbols s₀,...,s₃ is transmitted in T seconds. If symbol si is sent, a waveform x(t,si) = cos(2π f₀t + π/4 + iπ/2) is transmitted during the interval [0, T ]. In this example, the experiment is to transmit one symbol over [0, T] seconds and each sample function has duration T. In a real communications system, a symbol is transmitted every T seconds and an experiment is to transmit j symbols over [0, j T] seconds. In this case, an outcome corresponds to a sequence of j symbols and a sample function has duration j T seconds.