SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 251 
X(t) has a value in the infinitesimal range + < X(t) < x + dz at the 
time t. More generally, the probability that X(¢) lies in a finite range 
a < X(t) < b at the time ¢ is given by the integral 
Pla < o6@) <a = i ° fi(a,t) dx (10.1) 
The second-order probability-density function associated with a random 
variable X (¢) is designated f2(a1,t13;%2,t2) and is defined to be such that the 
product fo(#1,t1;%2,t2) dx: dx2 gives the probability that the variable lies in 
the range 21 < X(t1) < v1 + dz, at the time ¢, and also lies in the 
range %2 < X(te) < x2 + dxe at the time ¢,. Clearly the second-order 
density function provides a more detailed description of the signal than 
does the first-order function, and as higher-order density functions are 
prescribed more and more detail is available about X(¢). It is seldom 
indeed, however, that one is so fortunate as to know the probability- 
density functions of all orders which describe a random signal, and in 
many cases one must settle for much less specific information. 
In a typical situation one may know some of the moments of the dis- 
tributions which describe the random signal being considered. The first 
moment is the expected value or average or mean and is defined by the 
integral 
E[X(h)] = [%, afte) de (10.2) 
This moment depends upon the first-order density function, and gives, in 
a rough sense, the d-c content of the signal X(t). The generalized 
second moment of the random variable X(t) is the autocorrelation func- 
tion, defined by 
®,.(t1,t2) = X (t1).X (t2) = je ie XX of o(a1,t1 522, t2) dx, dx. (10.3) 
which is the average of the product of values of the variable at t; times the 
values of the variable at time t. For the special case when ¢; equals to, 
then x; equals x2, and 
®,.(t1,t1) 
[0 [0 eetilestiywats) des dar 
= les r17f1(x1,t1) dx 
E[X?(,)] (10.4) 
which is the mean-square value of the variable at t;. In the second step 
of (10.4), use has been made of the fact that the integral of the second- 
order function f2 over all values of x2 is just the first-order density func- 
tion fi. That is, the probability that the variable has a value in the 
range 21 < X < x + dz, at time ¢; and has any value at all at time f2 is 
simply fi(v1,t1) dvi. From the symmetry of (10.3) it is obvious that 
®,2(t1,t2) equals ®,,(t2,01). 
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