Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 169

As indicated in Sect. 2.7.2 , we now have a finite set of values x 1 i and x 2 j , and we
can calculate the arithmetic mean value of the products, also called the empirical
mean value , since it is obtained from the experiment,

P

N 1

P

N 2
j¼ 1 N ij x 1 i x 2 j
N

i¼ 1

X 1 X 2emp ¼ : (3.74)

For a high enough value N , the ratio N ij / N becomes a good approximation of the
probability,

N ij
N ! PfX 1 ¼ x 1 i ; X 2 ¼ x 2 j g;

(3.75)

and the empirical mean value becomes independent on experiment and approaches
the mean value of joint random variables X 1 and X 2 ,
X 1 X 2 ¼ X
i

X

x 1 i x 2 j PfX 1 ¼ x 1 i ; X 2 ¼ x 2 j g; (3.76)

j

or, in a general case,

X 1 X 2 ¼ 1
i¼1

1

x 1 i x 2 j PfX 1 ¼ x 1 i ; X 2 ¼ x 2 j g: (3.77)

j¼1

This result can be generalized for N discrete random variables,

X 1 ; ... ; X N ; (3.78)

as given in the following equation:

X 1 ... X N ¼ 1

1

x 1 i ... x Nj PfX 1 ¼ x 1 i ; ... ; X N ¼ x Nj g: (3.79)

i¼1 ...

j¼1

Similarly,

gðX 1 ... X N Þ¼ 1

1

gðx 1 i ... x Nj ÞPfX 1 ¼ x 1 i ; ... ;X N ¼ x Nj g: (3.80)

i¼1 ...

j¼1