Applied Statistics and Probability for Engineers

116 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Therefore,

and

Suppose a digital 1 is represented as a shift in the mean of the noise distribution to 1.8

volts. What is the probability that a digital 1 is not detected? Let the random variable Sdenote

the voltage when a digital 1 is transmitted. Then,

This probability can be interpreted as the probability of a missed signal.

EXAMPLE 4-16 The diameter of a shaft in an optical storage drive is normally distributed with mean 0.2508

inch and standard deviation 0.0005 inch. The specifications on the shaft are 0.2500 0.0015

inch. What proportion of shafts conforms to specifications?

Let Xdenote the shaft diameter in inches. The requested probability is shown in Fig. 4-18 and

Most of the nonconforming shafts are too large, because the process mean is located very near

to the upper specification limit. If the process is centered so that the process mean is equal to

the target value of 0.2500,

By recentering the process, the yield is increased to approximately 99.73%.

0.9973

0.998650.00135

P 1 Z 32 P 1 Z 32

P 1  3 Z 32

P 1 0.2485X0.2515 2 P a

0.24850.2500

0.0005

Z

0.25150.2500

0.0005

b

0.919240.00000.91924

P 1 4.6Z1.4 2 P 1 Z1.4 2 P 1 Z4.6 2

P 1 0.2485X0.2515 2 P a

0.24850.2508

0.0005

Z

0.25150.2508

0.0005

b

P 1 S0.9 2 P a

S1.8

0.45



0.91.8

0.45

bP 1 Z 22 0.02275

x2.58 1 0.45 2 1.16

x 0.452.58

0.2515

f(x)

0.2508

0.25

0.2485 x

Specifications

Figure 4-18

Distribution for

Example 4-16.

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