Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 165

Using ( 3.45 ), ( 3.46 ) is rewritten as:

Ð Ð

x 1 x 2 þ d x 2

f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2

1 x 2

F X 1 x 1 jx 2 < X 2 x 2 þ d x 2 ð Þ ¼

x 2 þ d x 2

Ð

f X 2 ðx 2 Þ d x 2

x 2

x 1

Ð

x 1

Ð

f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 f X 1 X 2 ðx 1 ; x 2 Þ d x 1

1 1

¼ ¼ :

f X 2 ðx 2 Þ d x 2 f X 2 ðx 2 Þ

(3.47)

If d x 2 is approaching zero, and for each x 2 for which f X 2 ðx 2 Þ 6¼ 0, we finally
obtain:

Ð

x 1

f X 1 X 2 ðx 1 ; x 2 Þ d x 1

1

F X 1 x 1 jX 2 ¼ x 2 ð Þ ¼ : (3.48)

f X 2 ðx 2 Þ

Similarly, we have:

x 2

Ð

f X 1 X 2 ðx 1 ; x 2 Þ d x 2

1

F X 2 x 2 jX 1 ¼ x 1 ð Þ ¼ : (3.49)

f X 1 ðx 1 Þ

From ( 3.48 ) and ( 3.49 ), using ( 3.27 ), we obtain the corresponding PDFs:

f X 1 X 2 ð x 1 ; x 2 Þ
f X 2 ðx 2 Þ

f X 1 ðx 1 jX 2 ¼ x 2 Þ¼ : (3.50)

f X 1 X 2 ð x 1 ; x 2 Þ
f X 1 ðx 1 Þ

f X 2 x 2 jX 1 ¼ x 1 ð Þ ¼ : (3.51)

Consider now that the condition for event B is defined as an event in which the
other variable X 2 lies in the given interval [ x 21 , x 22 ], resulting in:

Ð Ð

x 1 x 22

f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2

1 x 21

F X 1 x 1 jx 21 < X 2 x 22 ð Þ ¼

Ð

x 22

f X 2 ðx 2 Þ d x 2

x 21

F X 1 X 2 ð x 1 ; x 22 Þ F X 1 X 2 ð x 1 ; x 21 Þ
F X 2 ðx 22 ÞF X 2 ðx 21 Þ

¼ ;

(3.52)