In order to properly define what it means to be stationary
from a mathematical standpoint, one needs to be somewhat
familiar with the concepts of distribution and density
functions. If you can remember your statistics then feel free
to skip this section!
Recall that when dealing with a single random variable, the
probability distribution function is a simply
tool used to identify the probability that our observed random
variable will be less than or equal to a given number. More
precisely, let XX be our random
variable, and let xx be our given
value; from this we can define the distribution function as
F
x
x=PrX≤x
F
x
x
X
x
(1)
This same idea can be applied to instances where we have
multiple random variables as well. There may be situations
where we want to look at the probability of event
XX and
YY both occurring. For example,
below is an example of a second-order
joint distribution
function.
F
x
xy=PrX≤xY≤y
F
x
x
y
X
x
Y
y
(2)
While the distribution function provides us with a full view
of our variable or processes probability, it is not always the
most useful for calculations. Often times we will want to
look at its derivative, the probability density
function (pdf). We define the the pdf as
f
x
x=dd
x
F
x
x
f
x
x
x
F
x
x
(3)
f
x
xdx=Prx<X≤x+dx
f
x
x
dx
x
X
x
dx
(4) reveals some of the physical
significance of the density function. This equations tells
us the probability that our random variable falls within a
given interval can be approximated by
f
x
xdx
f
x
x
dx
. From the pdf, we can now use our knowledge of
integrals to evaluate probabilities from the above
approximation. Again we can also define a
joint density
function which will include multiple random variables
just as was done for the distribution function. The density
function is used for a variety of calculations, such as finding
the expected value or proving a random variable is stationary,
to name a few.
The above examples explain the distribution and density
functions in terms of a single random variable,
XX. When we are dealing with signals and
random processes, remember that we will have a set of random
variables where a different random variable will occur at each
time instance of the random process,
X
t
k
X
t
k
. In other words, the distribution and density
function will also need to take into account the choice of
time.
Autocorrelation Function
