Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 174

If the correlation is zero, the variables are said to be orthogonal ,

R X 1 X 2 ¼ EfX 1 ; X 2 g ¼ 0 : (3.112)

Note that if the variables are uncorrelated and one or both variables have a zero
mean value, it follows that they are also orthogonal.

Example 3.3.3 The random variables X 1 and X 2 are related in the following form:

X 2 ¼ 2 X 1 þ 5 : (3.113)

Determine whether or not the variables are correlated and orthogonal, if the
random variable X 1 has the mean, and the squared mean values equal to 2 and 5,
respectively.
Solution In order to determine if the variables are correlated and orthogonal, we
first have to find the correlation:

R X 1 X 2 ¼ EfX 1 X 2 g ¼ E X 1 ð 2 X 1 þ 5 f g ¼ Ef 2 X 1 þ 5 X 1 g
¼ 2 EfX 1 gþ 5 EfX 1 g ¼ 2 5 þ 5 2 ¼ 0

: (3.114)

Therefore, according to ( 3.112 ) the variables are orthogonal.
Next, we have to verify that the condition ( 3.111 ) is satisfied. To this end, we
have to find the mean value of X 2 , being as the mean value of X 1 has been found to
be equal to 2.

EfX 2 g ¼ 2 EfX 1 gþ 5 ¼ 4 þ 5 ¼ 1 : (3.115)

Since

EfX 1 X 2 g ¼ R X 1 X 2 ¼ 0 6¼ EfX 1 gEfX 2 g ¼ 2 1 ¼ 2 ; (3.116)

the variables are correlated.

3.3.3 Joint Central Moments

The joint central moment of order r , of two random variables X 1 and X 2 , with
corresponding mean values E { X 1 } and E { X 2 }, is defined as:

n o

n X 2 EfX 2 g k

m nk ¼ E X 1 EfX 1 g ð Þ ð Þ ; (3.117)

where the order r is:

r ¼ n þ k: (3.118)