where P {& <x) is the probability that § is less than or equal to x. 
The distribution function is nonnegative, nondecreasing, and continuous 
from the left; also F(- ~) = O and F(™) = 1. 
Analogously, if € is an n-truple of random variables, its distri- 
bution function is a function of n real variables. 
Ruste = < 6.00 
F(x); Xo» > x) iP {E as Bae 3 E < x} 
and F is called a joint distribution function of the variables Ke The 
function F(x), Koorees x) is uniquely defined in n-dimensional Euclidian 
space ED is non-decreasing, and is continuous from the left with respect 
to each variable. Furthermore, 
F(x), Kore Kin 7 Ms Xe pooees 
and 
Cope pape te eae) r= gl) Ge 
where po) denotes the distribution function of the i-truple (Eporee> E5)- 
A random function or a stochastic process is a random variable &(t) 
which is a function of time. As time varies, &(t) describes the evolu- 
tion of the process. If a random process is recorded as it evolves, the 
recorded function €(*) describes only one of the many possible ways in 
which the process might have developed. The recorded function &(°) is 
called a sample function of the random process. For each fixed value of 
t, the quantity §(t) is a random variable. 
Whereas a random variable is characterized by a distribution function, 
a stochastic process is characterized by a set of joint distribution 
functions. Assume that it is possible to assign a probability distribution 
to the multidimensional random variable ee = (E(t,)> E(ty)s--es g(t )) 
for any n and arbitrary times t.. The distribution function 
43 
