Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 162

From ( 3.22 ), we get:

f X 1 X 2 ðx 1 ; x 2 Þ¼ 1 = 2 dðx 1 1 Þdðx 2 1 Þþ 1 = 4 dðx 1 1 Þdðx 2 Þþ 1 = 4 dðx 1 Þdðx 2 1 Þ:

(3.31)

Next, in Fig. 3.7 we compare one-dimensional and two-dimensional PDFs.
The shaded area in Fig. 3.7a represents the probability that the random variable
X is in the infinitesimal interval [ x , x +d x ]:

A ¼ Px< X x þ d x f g: (3.32)

From here, considering that d x is an infinitesimal interval, the PDF in the interval
[ x , x +d x ] is constant, resulting in:

f X ðxÞ d x ¼ A ¼ Px< X x þ d x f g: (3.33)

Similarly, the volume in Fig. 3.7b presents the probability that the random
variables X 1 and X 2 are in the intervals [ x 1 , x 1 +d x 1 ] and [ x 2 , x 2 +d x 2 ],

respectively.
The equivalent probability

Px 1 < X 1 x 1 þ d x 1 ; x 2 < X 2 x 2 þ d x 2 f g (3.34)

corresponds to the elemental volume V , with a base of (d x 1 d x 2 ) and height of
f X 1 X 2 ðx 1 ; x 2 Þ :

f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 ¼ V ¼ Px 1 < X 1 x 1 þ d x 1 ; x 2 < X 2 x 2 þ d x 2 f g: (3.35)

Fig. 3.7 One-dimensional and two-dimensional PDFs