Applied Statistics and Probability for Engineers

120 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

EXAMPLE 4-18 The digital communication problem in the previous example is solved as follows:

Because and n(1 p) is much larger, the approximation

is expected to work well in this case.

EXAMPLE 4-19 Again consider the transmission of bits in Example 4-18. To judge how well the normal

approximation works, assume only n50 bits are to be transmitted and that the probability

of an error is p0.1. The exact probability that 2 or less errors occur is

Based on the normal approximation

Even for a sample as small as 50 bits, the normal approximation is reasonable.

If npor n(1 p) is small, the binomial distribution is quite skewed and the symmetric

normal distribution is not a good approximation. Two cases are illustrated in Fig. 4-20.

However, a correction factor can be used that will further improve the approximation. This

factor is called a continuity correctionand it is discussed in Section 4-8 on the CD.

P 1 X 22 P a

X 5

2.12



2  5

2.12

bP 1 Z1.42 2 0.08

P 1 X 22  a

50

0

b 0.9^50 a

50

1

b 0.1 1 0.9^492 a

50

2 b^ 0.1

(^21) 0.9 (^482) 0.112
np 116  106211  10 ^52  160
P 1 Z0.79 2 P 1 Z0.79 2 0.785
P 1 X 1502 P a
X 160
216011  10 ^52

150  160
216011  10 ^52
b
Figure 4-20 Binomial
distribution is not
symmetrical if pis near
0 or 1.
012345678910
0.0
0.1
0.3
0.4
x
f(x)
0.2
np
10 0.1
10 0.9
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