563489578934

B–7 Examples of Important Distributions 699
statistics, and many theorems have been developed by statisticians that are based on
Gaussian assumptions. It cannot be overemphasized that the Gaussian distribution is very
important in analyzing both communication problems and problems in statistics. It can also
be shown that the Gaussian distribution can be obtained as the limiting form of the binomial
distribution when n becomes large, while holding the mean m = np finite, and letting the
variance s 2 = np(1 - p) be much larger than unity [Feller, 1957; Papoulis, 1984].
DEFINITION. The Gaussian distribution is
1
>(2s
f(x) =
2 )
(B–45)
12ps e-(x-m)2
where m is the mean and s 2 is the variance.
A sketch of Eq. (B–45) is given in Fig. B–5, together with the CDF for the Gaussian
random variable. The Gaussian PDF is symmetrical about x = m, with the area under the PDF
1
1
being
2
for (-q … x … m) and
2
for (m … x … q). The peak value of the PDF is 1>112ps2,
so that as s S 0, the Gaussian PDF goes into a δ function located at x = m (since the area
under the PDF is always unity).
We will now show that Eq. (B–45) is properly normalized, (i.e., that the area under f(x)
is unity). This can be accomplished by letting I represent the integral of the PDF:
q
q
1
>(2s
I ! f(x)dx = 2 ) dx
(B–46)
L -q L -q 12p s e-(x-m)2
Making a change in variable, we let y = (x - m)/s. Then,
q
q
1
I =
e -y2 >2 1
(sdy) = e -y2 >2 dy
(B–47)
12p s L-q
12p L-q
The integral I can be shown to be unity by showing that I 2 is unity:
q
q
I 2 1
= c e -x2 >2 1
dx dc e -y2 >2
dy d
12p L-q
12p L-q
= 1 q q
e -(x2 +y 2 )>2
dx dy
(B–48)
2p L-q L-q
Make a change in the variables to polar coordinates from Cartesian coordinates. Let r 2 = x 2 + y 2 ,
and let u = tan -1 (y>x); then,
I 2 = 1 2p q
c e -r2 >2 r dr d du = 1 2p
du = 1 (B–49)
2p L0
L 0
2p L0
Thus, I 2 = 1, and, consequently, I = 1.
Up to now, we have assumed that the parameters m and s
2 of Eq. (B–45) were the mean
and the variance of the distribution. We need to show that indeed they are! This may be
accomplished by writing the Gaussian form in terms of some arbitrary parameters a and b:
f(x) =
1
12p b e-(x-a)2 >(2b 2 )
(B–50)