Page 75 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

Note that by setting a = ka, in Eq. (2.97), we obtain

Equation (2.98) says that the probability that a r.v. will fall k or more standard deviations from its mean is
< l/k2. Notice that nothing at all is said about the distribution function of X. The Chebyshev inequality is
therefore quite a generalized statement. However, when applied to a particular case, it may be quite weak.

SPECIAL DISTRIBUTIONS
2.37. A binary source generates digits 1 and 0 randomly with probabilities 0.6 and 0.4, respectively.
(a) What is the probability that two 1s and three 0s will occur in a five-digit sequence?
(b) What is the probability that at least three 1s will occur in a five-digit sequence?
(a) Let X be the r.v. denoting the number of 1s generated in a five-digit sequence. Since there are only two
possible outcomes (1 or O), the probability of generating 1 is constant, and there are five digits, it is
clear that X is a binomial r.v. with parameters (n, p) = (5, 0.6). Hence, by Eq. (2.36), the probability
that two 1s and three 0s will occur in a five-digit sequence is

(b) The probability that at least three 1s will occur in a five-digit sequence is

where

Hence, P(X 2 3) = 1 - 0.317 = 0.683

2.38, A fair coin is flipped 10 times. Find the probability of the occurrence of 5 or 6 heads.
Let the r.v. X denote the number of heads occurring when ia fair coin is flipped 10 times. Then X is a
binomial r.v. with parameters (n, p) = (10, 4). Thus, by Eq. (2.36),

2.39. Let X be a binomial r.v. with parameters (n, p), where 0 <: p < 1. Show that as k goes from 0 to
n, the pmf p,(k) of X first increases monotonically and then decreases monotonically, reaching its
largest value when k is the largest integer less than or equal to (n + 1)p.
By Eq. (2.36), we have

Hence, px(k) 2 px(k - 1) if and only if (n - k + l)p 2 k(l - p) or k I (n + 1)p. Thus, we see that px(k)
increases monotonically and reaches its maximum when k is the largest integer less than or equal to
(n + 1)p and then decreases monotonically.