Random Variable - Random Signals and Processes Primer with MATLAB - page 75

From ( 2.200 ) and ( 2.202 ), we have:
f Y ðyÞ¼ X
i

Pfy i 0 gdðy y i 0 Þ¼ 0 : 9 dðy 2 Þþ 0 : 1 dðy 1 Þ;

F Y ðyÞ¼ X
i

(2.203)

Pfy i 0 guðy y i 0 Þ¼ 0 : 9 uðy 2 Þþuðy 1 Þ:

2.7 Mean Value

The probability density and distribution functions provide all of the information
about the particular random variable being considered. However, there are many
applications in which we do not need that much information but where we need
some parameters that are representative of a particular distribution without finding
the entire density or distribution functions [HAD06, p. 110], [NGU09, p. 86].
The mean value of a random variable is one of the single most important parameters
associated with a random variable [PAP65, p. 138], [BRE69, p. 56]. The mean
value plays an important role in the characterization of the random variable when
a partial description is either needed or only possible.

2.7.1 What Is a Mean Value?

We are all familiar with the term “average” or “mean value” when referring to a
finite known set of values. For example, an average grade of an exam, average
salary, an average monthly temperature in a town, etc. As an example, denote the
temperature measured in i th day as t i , then the average temperature t av is obtained
by adding all t i and dividing by the number of the days

t 1 þ t 2 þþ t 30
30

t av ¼ : (2.204)

The temperature t av can be viewed as the “most likely” or “expected” temperature
in a month, yet may never happen. For example, if the average temperature calcu-
lated in ( 2.204 ) is 16.5 C, it may never occur that the measured temperature was
16.5 C. However, generally speaking we have enough information about the
weather during this month (i.e., we do not expect snow, but we are also not expecting
very hot weather). Similarly, an average salary in a country tells us about the
standard of the life in this country; the average grade on an exam tells us about
the general success of students on this exam; an average speed of 110 km/h in a car
tells that the driver drives very fast, etc.
Generally, when a large collection of numbers is assembled, we are not interested
in the individual numbers, but rather in the average value. The average temperature,