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B–7 Examples of Important Distributions 695
Thus, there are two ways to evaluate the variance: (1) by use of the definition as given
by Eq. (B–27) and (2) by use of the theorem that is Eq. (B–29).
B–7 EXAMPLES OF IMPORTANT DISTRIBUTIONS
There are numerous types of distributions. Some of the more important ones used in communication
and statistical problems are summarized in Table B–1. Here, the equations for their PDF
and CDF, a sketch of the PDF, and the formula for the mean and variance are given. These
distributions will be studied in more detail in the paragraphs that follow.
Binomial Distribution
The binomial distribution is useful for describing digital, as well as other statistical problems.
Its application is best illustrated by an example.
Assume that we have a binary word n bits long and that the probability of sending a
binary 1 is p. Consequently, the probability of sending a binary 0 is 1 - p. We want to evaluate
the probability of obtaining n-bit words that contain k binary 1s. One such word is k binary
1s followed by n - k binary 0s. The probability of obtaining this word is p k (1 - p) n-k . There
are also other n-bit words that contain k binary 1s. In fact, the number of different n-bit words
containing k binary 1s is
a n (B–32)
k b = n!
(n - k)!k!
(This can be demonstrated by taking a numerical example, such as n = 8 and k = 3.) The
symbol Ak n B is used in algebra to denote the operation described by Eq. (B–32) and is read “the
combination of n things taken k at a time.” Thus, the probability of obtaining an n-bit word
containing k binary 1s is
P(k) = a n k bpk (1 - p) n-k
(B–33)
If we let the random variable x denote these discrete values, then x = k, where k can take on
the values 0, 1, 2,..., n, and we obtain the binomial PDF
n
f(x) = a P(k)d(x - k)
k = 0
(B–34)
where P(k) is given by Eq. (B–33).
The name binomial comes from the fact that the P(k) are the individual terms in a
binomial expansion. That is, letting q = 1 - p, we get
(p + q) n n
= a a n k b pk q n-k n
= a P(k)
k = 0
k = 0
(B–35)