that your values fall within any range of possible values. A Review of Some Important Math, Stats, and Systems 3.2 Random Processes 3.2.1 A Definition Random processes are sort of like random variables, only there's a little more to know. We'll work our way to defining a random process like this: (1) First, let's say we have A, a random event (an event with an unknown out- come). We'll make A the random event: "Will it be sunny tomorrow?" (2) Now, a random variable X is a x(t,A1) number whose exact value de- pends on the random event A. For example, X may be tomorrow's high temperature, and that will depend on whether it's sunny or A = A1 not. So we can write X = = (x(A)- that is, X is a function of A. (3) Now, we jump to a random process. A random process, x(t), is a function of time t, where the exact function that occurs de- x(t,A2) pends on a random event A. For example, let x(t) be tomorrow's temperature as it changes with time over the day; the values of x(t) will depend on A (whether A = A2 it's sunny or not). So, we can write x (t) = x (t,A) to indicate that the time function depends on the random event A. Here, X (t,A) is a random process. A plot of a random process x(t,AN) is shown in Figure 3.3. We see that there is a time function x(t) = (t,A), and the exact value of the time function x(t) = x (t,A) de- pends on the random event A. If Figure 3.3 Plot of a random process x(t,A) Chapter Three the random event A corresponds to its first possible outcome-that is, it is sunny tomorrow-then we get the time function x (t,A) telling us tomorrow's temperature throughout the day. If the random event A corresponds to its second possible outcome, A2-it is partly cloudy tomorrow-then we get time function X (t,A2) telling us how tomorrow's temperature changes throughout the day. And SO on. There's one very important thing to note about a random process x (t,A). Take a look at Figure 3.4. There I've drawn (t,A) and I've drawn a dotted line through time t=t1. = At time t=t1, we have x (t1,A), which is a number whose 