B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

8.2 Random Variables 423
Figure 8.13
(a) Joint PDF.
(b) Conditional
PDF.
Volume = Pxy(x,y) dx dy
(b)
The event of observing x in the interval ( -oo, oo) and observing yin the interval ( -oo, oo)
is a certainty. Hence,
1_: 1_: Pxy (x, y) dxdy = I
(8.47)
Thus, the total volume under the joint PDF must be unity.
When we are dealing with two RVs x and y, the individual probability densities P x (x)
and Py (y) can be obtained from the joint density P xy (x, y). These individual densities are also
called marginal densities. To obtain these densities, we note that P x (x) x is the probability
of observing x in the interval (x, x + x). The value of y may lie anywhere in the interval
(-oo, oo). Hence,
lim P x (x)x = lim Probability (x < x :S x + x, -oo < y :S oo)
L'u-0
f>x-0
1 x+ t>
x co
!
6-x--,,0
!
x -co
co x+ t> x
f>x-0
f
- CO
X
= lim
= lim
Pxy (x, y) dxdy
P xy (x, y) dy dx
= lim x f co P xy (x, y) dy
lu->0 -co