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698
Probability and Random Variables
Appendix B
Recalling that p and q are probabilities and p + q = 1, we see that (p + q) n-1 = 1. Thus,
Eq. (B–39) reduces to
m = np
(B–40)
Similarly, it can be shown that the variance is np(1 - p) by using s 2 = x 2 - (xq) 2 .
Poisson Distribution
The Poisson distribution (Table B–1) is obtained as a limiting approximation of a binomial
distribution when n is very large and p is very small, but the product np = l is some reasonable
size [Thomas, 1969].
Uniform Distribution
The uniform distribution is
0, x 6 a 2m - A b
2
f(x) = g
1
A , |x - m| … A 2
(B–41)
0, x 7 a 2m + A b
2
where A is the peak-to-peak value of the random variable. This is illustrated by the sketch
shown in Table B–1. The mean of this distribution is
and the variance is
L
Making a change in variable, let y = x - m:
(B–42)
(B–43)
s 2 = 1 y 2 dy = A2
(B–44)
A L-A>2
12
The uniform distribution is useful in describing quantizing noise that is created when an
analog signal is converted into a PCM signal, as discussed in Chapter 3. In Chapter 6, it is
shown that it also describes the noise out of a phase detector when the input is Gaussian noise
(as described subsequently).
Gaussian Distribution
q
-q
m+(A>2)
x f(x) dx = x 1
L A dx = m
m+(A>2)
s 2 = (x - m) 2 1
L A dx
m-(A>2)
m-(A>2)
A>2
The Gaussian distribution, which is also called the normal distribution, is one of the most
important distributions, if not the most important. As discussed in Chapter 5, thermal noise
has a Gaussian distribution. Numerous other phenomena can also be described by Gaussian