S ax,ay À 
points, 
5.5 
variable. Random processes arise in many problems of practical 
interest: by a random function we can describe the time 
fluctuations of the refractive index at a spatial point, the 
fiim deformation due to the photographic processing along one 
longituainal section (here t stands for length) and so on. All 
these examples have in common, that for a fixed value of the 
argument the values of the corresponding functions are ranaom 
variables. As in many problems of practical importance t stands 
for time, we shall cali t the time. 
Let us now consider how we can mathematically specify such a 
random function. Let us fix the value of the argument t of the 
random function, putting for example t=t,. Then, if we are only 
4 
interested in the value of the function X(t) at that time 
instant t it is sufficient to give the density distribution 
1? 
of the random variable X= X(t,). This density distribution 
can however not give a complete description of the random 
function, because it does not express the mutual dependency 
of the ordinates for different values of t. In order to obtain 
a more detailed description, let us take two values of the 
argument t, fer example t=t, and t=t,. The ordinates 
corresponding to them will be the mutually dependent random 
variables X, = X(t,) and X, 
characterized by the corresponding two-dimensional density 
= X(t,)» which can be completely 
distribution of Lo and X. This process can be carried further, 
by introducing new values t t, «.. and three-dimensional, 
3: 4 
four-dimensional density distributions and so on. 
The random function can be considered to be defined, if all the 
multi-dimensional density functions are given for any values t 
t t t 
2**** fü 
However, this method of defining the function is very cumbersome 
4? 
and it is more convenient to limit our studies to some 
characteristic parameters of these distributions. Let us consider 
the mathematical expectation (mean value) of the ordinate X 
for the value t = t. 
X(t,) = E(X(t,)) 
As a similar expression is valid for any other value of t, 
we may write 
X(t) = E(X(t))