Wiley signals and systems e book TLFe BO 422

17.2. Expected Values

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407

Changing statistical properties over time is not a, concern for normal dice.
We can further say that the average for niariy dice at the snme time will yield
approxirnately the same results as one die thrown repeatedly. The numbers
that, come up are the discrete values of a sample function along the time axis.
In this ease, the expected value can also be expressed 135; the t,ime-a;verage.

Before proceeding with the relationships betwccn the ensemble-avexage arid
tirne-average, we will discuss sonic more general forms of expected valiies.

The expected value E { z ( t ) )tells us what value to expect on average from a random
process, but it does not fully cliaracterist. the process.
Figure 17.2 shows sample ftmclioiis of two random processes that have the
same ( t ~ ~ ~ e - average,
v ~ r ~but
~ ~they
~ eclearly
~
differ in otlier € ) ~ ~ ) ~ e rThe
~~es.
sample functions of random process B vary much more around the axerage than
those of process A. In order to describe such properties we introduce the general
,first-ordcr expecled vulue:
(17.3)

In contrast to (17.1), z(L) is here replaced by a function J(n:(t)). By choosing

different fnnctions f , different first-order averages are obtained. The reason why we
use the term first-order expected values is because they oniy take into account the
a ~ ~ ~ i t uofd the
e s sample f u n c t ~ at
~ ~one
~ spoint in time. We will soon feasn about
higher-order expected values, where Glie values at,more than one time are combined
together. The mean according to (17.1) is contained in (17.3) for f(x) = 2. It is
also cdled the Eineur avemge and is denoted by E(z(t)}= pz(t).
For f i x ) = x2 we obtain the ~ ~ a ~ u~7 ~ue ~~~ , gze :c
(17.4)
M’c can use it to describe the average power of a random process, for example, the
noise power of an amplifier witli~~ut
lo,?xl.
The squaw of the deviation from the fillear average is also important. It is
obtained fronr (17.3) for f(.r) = (z - p . s ) 2and is called the uarmncc:

(17.5)