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

8.2 Random Variables 417
This integral cannot be evaluated in a closed form and must be computed numerically. It is
convenient to use the function Q(.), defined as 2
1 loo 2
Q(y) £ -- e-x 1 2 dx
(8.35)
,Jiii y
The area underpx (x) from y to oo (shaded in Fig. 8.10a) is* Q(y). From the symmetry of Px(x)
about the origin, and the fact that the total area under Px (x) = 1, it follows that
Q(-y) = 1 - Q(y)
(8.36)
Observe that for the PDF in Fig. 8. 10a, the CDF is given by (Fig. 8.10c)
Fx (x) = 1 - Q(x)
(8.37)
The function Q(x) is tabulated in Table 8.2 (see also later: Fig. 8. 12d). This function is widely
tabulated and can be found in most of the standard mathematical tables. 2 • 3 It can be shown
that, 4 1 2
Q(x) :::'. -- e-x / 2
x,Jiii
for x » l
(8.38a)
For example, when x = 2, the error in this approximation is 18.7%. But for x = 4 it is 10.4%
and for x = 6 it is 2.3%.
A much better approximation to Q(x) is
1 0.7 2;2
Q(x) :::'. -- ( 1 - - ) e-x
x,Jiii x 2 X>2 (8.38b)
The error in this approximation is just within 1 % for x > 2.15. For larger values of x the error
approaches 0.
A more general Gaussian density function has two parameters (m, a) and is (Fig. 8.11)
Px (x) = _l _ e- ( x-
m}2 /2 a
2
a$
(8.39)
For this case,
1 fx 2 2
Fx (x) = --
e-( x-
m ) /2 a dx
a$ - oo
* The function Q(x) is closely related to functions erf (x) and erfc (x),
2 roo 2
erfc (x) = ,Jrr x e-y dy = 2Q(x✓2)
J
Therefore,
Q(x) = erfc () = [1 -erf ()]