Random Variable - Random Signals and Processes Primer with MATLAB - page 78

The subscript “emp” indicates that the mean value is obtained in the experiment.
This mean value is called the empirical mean value .
If we repeat the experiment again, we will obtain another empirical mean value.
All of these mean values depend on the corresponding experiment (i.e., the average
value of the outcomes is not predictable).
Example 2.7.1 A die rolling experiment is performed three times. In each experi-
ment, the die is rolled 100 times and the number of dots facing up is counted and
noted. The results are shown in Table 2.3 .
The mean value obtained in the first experiment is:

1 18 þ 2 20 þ 3 17 þ 4 15 þ 5 14 þ 6 16
100

m emp 1 ¼ ¼ 3 : 35 :

(2.215)

Similarly, in the second and third experiments, we have:

1 13 þ 2 22 þ 3 18 þ 4 15 þ 5 17 þ 6 15
100

m emp 2 ¼ ¼ 3 : 46 :

(2.216)

1 12 þ 2 16 þ 3 19 þ 4 14 þ 5 16 þ 6 18
100

m emp 3 ¼ ¼ 3 : 45 :

(2.217)

The obtained mean values are different. Similarly, in each experiment, we will
obtain a different empirical mean value.
However, we are interested in a mean value which does not depend on the
experiment. Intuitively, we can say that the empirical mean values will vary a little
if the number of repetitions of each experiment M is very high ( M >>

1).
To this end, note that the ratio N i / M in ( 2.214 ) represents the relative frequency
which approaches the probability of the event x i ,if M 1 (see ( 1.38 )).

Table 2.3 The results obtained in three experiments
Experiment 1 Experiment 2 Experiment 3
Number of dots Occurrence Number of dots Occurrence Number of dots Occurrence
1 18 1 13 1 12

2 20 2 22 2 16

3 17 3 18 3 19

4 15 4 15 4 14

5 14 5 17 5 16

6 16 6 15 6 18