exact value depends on A. That's just a random variable! So, the sample of a x(t,A1) random process X (t,A) at = is a random variable. We'll call it X1=x1(A). A = A1 x(t1, A1) x(t,A2) A = A2 x(t1, A2) x(t,AN) Figure 3.4 A random process, highlighting time t = t1 x(t1, AN) A Review of Some Important Math, Stats, and Systems 3.2.2 Expressing Yourself, or a Complete Statistical Description Now, you say, I understand what a random process is. It's simply a function of time whose exact function depends on a random event A. But how do I characterize it? Let's say I want to tell my mathematician friends all about my random process-how do I do that? To totally and fully describe your random process, you'll have to provide this information: (1) At any time t = t, you'll have (t,A) = x(t,,A) = X1, a random variable. Your mathematically inclined friends will need to know all about this random variable X1, which means you'll want to provide p(x). (2) At any two times t=t, and = t=t,, = you'll have x (t1,A) = X1 and x(t,,A) = X2, both of which are random variables. You will need a complete characterization of these two random variables, and as you know, that's done by providing the joint prob- ability density function p(x1,x2). (3) At any K times, t=t,t=t,, and SO on up to t = tk you'll have x (t1,A) = X1, (t,,A) = X2 and SO on up to x(t,A) = XK, all of which are random variables. This time, you'll have to provide a complete characterization of these random vari- ables, which means you'll have to provide Actually, all you have to give your friends is what's in (3), since you can use p(x1,X23---,XXX) to get the information in (1) and (2), namely p(x) and p(x),x2). However, providing the information in (3) is often tough stuff. Generally, it's difficult to provide for a random process. 3.2.3 Expressing Some of Yourself, or a Partial Description What most people do is provide partial information about a random process-just enough information to be able to proceed with a telecommunications study. What they provide is called a sec