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

436 FUNDAMENTALS OF PROBABILITY THEORY
ax or a; is often used as a measure of the width of a PDF. In Chapter 10, we shall use this
measure to estimate the bandwidth of a signal spectrum. The proof of Eq. (8.74) is as follows:
Because the integrand is positive,
If we replace x by its smallest value ka x , the inequality still holds,
or
Hence,
1
P(lxl < kax) 1 - k
2
This inequality can be generalized for a nonzero mean RV as:
- 1
P(l x - xi < kax) 1 - k
2
(8.75)
Example 8.23 Estimate the width, or spread, of a Gaussian PDF [Eq. (8.60a)]
For a Gaussian RV [see Eqs. (8.35) and (8.40b)]
P(l x - xi < a) = 1 - 2Q(l) = 0.6826
P(lx - xi < 2a) = 1 - 2Q(2) = 0.9546
P(l x - xi < 3a) = 1 - 2Q(3) = 0.9974
This means that the area under the PDF over the interval (x -3a, x +3a) is 99.74% of the
total area. A negligible fraction (0.26%) of the area lies outside this interval. Hence, the
width, or spread, of the Gaussian PDF may be considered roughly ±3a about its mean,
giving a total width of roughly 6a.
8.4 CORRELATION
Often we are interested in determining the nature of dependence between two entities, such
as smoking and lung cancer. Consider a random experiment with two outcomes described by