B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

8.4 Correlation 437
RVs x and y. We conduct several trials of this experiment and record values of x and y for each
trial. From this data, it may be possible to determine the nature of a dependence between x
and y. The covariance of RVs x and y is one measure that is simple to compute and can yield
useful information about the dependence between x and y.
The covariance CJ xy of two RVs is defined as
Clxy (x - x)(y - y) (8.76)
Note that the concept of covariance is a natural extension of the concept of variance, which is
defined as
(J; = (x - x)(x - x)
Let us consider a case of two variables x and y that are dependent such that they tend
to vary in harmony; that is, if x increases y increases, and if x decreases y also decreases.
For instance, x may be the average daily temperature of a city and y the volume of soft drink
sales that day in the city. It is reasonable to expect the two quantities to vary in harmony for a
majority of the cases. Suppose we consider the following experiment: pick a random day and
record the average temperature of that day as the value of x and the soft drink sales volume
that day as the value of y. We perform this measurement over several days (several trials of
the experiment) and record the data x and y for each trial. We now plot points (x, y) for all
the trials. This plot, known as the scatter diagram, may appear as shown in Fig. 8.1 8a. The
plot shows that when x is large, y is likely to be large. Note the use of the word likely. It is not
always true that y will be large if x is large, but it is true most of the time. In other words, in
a few cases, a low average temperature will be paired with higher soft drink sales owing to
some atypical situation, such as a major soccer match. This is quite obvious from the scatter
diagram in Fig. 8.1 8a.
To continue this example, the variable x - x represents the difference between actual
and average values of x, and y - y represents the difference between actual and average
values of y. It is more instructive to plot (y - y) vs. (x - x). This is the same as the scatter
diagram in Fig. 8.18a with the origin shifted to (:x, y), as in Fig. 8.18b, which shows that a day
with an above-average temperature is likely to produce above-average soft drink sales, and
a day with a below-average temperature is likely to produce below-average soft drink sales.
Figure 8.18
Scatter
diagrams:
(a), (b) positive
correlation;
(c) negative
correlation;
(d) zero
correlation.
y t
.. ..
.. . ·.::.·-: ·
, ·. ...
:·•
. .. -
(a)
x-
(y - y) t
(z - z) t (w - w) t
-
:=-=:· (x - x)
(b)
.. .
: .
..·. : ·: .
:
(x - x) (x - x)
_
(c) (d)