Applied Statistics and Probability for Engineers

Therefore, (x10) 2 2.05, and the standardizing transformation is used in reverse to solve

for x. The result is

EXAMPLE 4-15 Assume that in the detection of a digital signal the background noise follows a normal distri-

bution with a mean of 0 volt and standard deviation of 0.45 volt. The system assumes a digi-

tal 1 has been transmitted when the voltage exceeds 0.9. What is the probability of detecting

a digital 1 when none was sent?

Let the random variable Ndenote the voltage of noise. The requested probability is

This probability can be described as the probability of a false detection.

Determine symmetric bounds about 0 that include 99% of all noise readings. The question

requires us to find xsuch that. A graph is shown in Fig. 4-17. Now,

From Appendix Table II

P 1 2.58Z2.58 2 0.99

P 1 x 0.45Zx 0.45 2 0.99

P 1 xNx 2 P 1 x 0.45N 0.45x 0.45 2

P 1 xNx 2 0.99

P 1 N0.9 2 P a

N

0.45



0.9

0.45

bP 1 Z 22  1 0.977250.02275

x 21 2.05 2

10 14.1 milliamperes

4-6 NORMAL DISTRIBUTION 115

10 x

z =x –^10 = 2.05

2

Figure 4-16 Deter- 0.98

mining the value of x

to meet a specified

probability.

Standardized distribution of0.45N

• z 0 z –x 0 x

Distribution of N

Figure 4-17 Deter-

mining the value of x

to meet a specified

probability.

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