Wiley signals and systems e book TLFe BO 421

406

17. Describing Random Signals
I

I

I

0

e

I
I

Figure 17.1: Example of sample functiurls

6

2 ,( t )

Under certain pre-conditions, that we will deal with later in niorc dctail, the
enscnible average can also be expressed by the time-average for a sample
function.

Example 17.1

We use the example of a die to show the three methods for finding an exact ox
approximate expected value:
6

Fkom howledge of the process, which is the complete symirictry of the die,
we can say that every number has the same prob;thility of appearing, so thc
expectcd v:tlur is:

E { z ( t ) }=

14-2 + 3 + 4 + 5 + 6
= 3.5 *
6

As the die's syrrirrietry does riot change, the expected valiie
any time t.
e

is

the same for

If wc repeat the procedure with ~ n a n ydice whose syrnrnetxy properties we
do riot know precisely, wc can also approximately calculate the eiisenible
average by averaging all of the iiurrhm on tlie die's faces. As the expected
values can in general change over time, we should take the ensemble Iiiean
for all dice at, the sairie time.