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p128_exercises_4-6

p128_exercises_4-6 - 128 CHAPTER 4 CONTINUOUS RANDOM...

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Unformatted text preview: 128 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Figure 4. 18 Distribution for Example 4-16. EXAMPLE 4—16 Shaft Diameter f(x) 1* Specifications x , .‘ ... .....__1-MJ...._.....W _._L ....W .. _ _ a, 0.2485 / 0.2508 0.2515 x 0.25 r..._.-,w._ _._ . and x = 2.58(0.45) = 1.16 Suppose that when a digital 1 signal is transmitted the mean of the noise distribution shifts to 1.8 volts. What is the probability that a digital 1 is not detected? Let the random variable S denote the ‘ voltage when a digital 1 is transmitted. Then, S — 1.8 0.9 — 1.8 < —. 0.45 0.45 P(S < 0.9) = P< ) = P(Z < —2) = 0.02275 { This probability can be interpreted as the probability of a missed signal. The diameter of a shalt 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 1 0.0015 inch. What propor- tion of shafts conforms to speciﬁcations? Let X denote the shaft diameter in inches. The requested probability is shown in Fig. 4-18 and . 85 — .25 .251 — 0.2 P(0.2485 < X < 0.2515) = P (w M) < z < 0.0005 0.0005 = P(—4.6 < Z < 1.4) = P(Z < 1.4) — P(Z < —4.6) = 0.91924 — 0.0000 = 0.91924 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, . < < . = < P(02485 X 02515) P< 0.0005 00005 = P(~3 < Z < 3) = P(Z < 3) — P(Z < —3) = 0.99865 — 0.00135 = 0.9973 0.2485 — 0.2500 0.2515 — 0.2500) ‘ Z < * f. By recentering the process, the yield is increased to approximately 99.73%. EXERCISES FOR SECTION 4—6 a 4—41. Use Appendix Table III to determine the following 4—42. Use Appendix Table III to determine the following (a) P(Z < 1.32) probabilities for the standard normal random variable Z: probabilities for the standard normal random variable Z: (b) P(Z < 3.0) (a) P(—-1 < z< 1) (b) P(—2 < z < 2) (d) P(Z > —2.15) (c) P(—3 < z < 3) (d) P(Z > 3) (c) P(Z > 1.45) (e) P(—2.34 < Z < 1.76) (e) P(O < Z< 1) ...
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