Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 179

Fig. 3.10 Positive and negative correlations

Fig. 3.11 Dependent uncorrelated and independent variables

If the random variables are independent, then obviously there is no relation
between them (not even a linear relation), resulting in a zero value of the coefficient
of correlation, as shown in Fig. 3.11b .
From ( 3.135 )to( 3.137 ), the values of the correlation coefficient are:

1 r X 1 X 2 1 : (3.138)

Example 3.3.4 Consider dependent random variables X and Y , where

Y ¼ X 2

: (3.139)

Determine whether or not the random variables X and Y are correlated and find
the coefficient of correlation for the following two cases:

(a) The random variable X is uniform in the interval [ 1, 1].
(b) The random variable X is uniform in the interval [0, 2].