chugg_mt1-94_sol

chugg_mt1-94_sol - that 1-e-λ b T min = 0 . 99 ⇐⇒ T...

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bad computer good computer 0.01 0.99 rejected accepted P D P FA P D 1 - 1 - P FA 1 EE464 - Chugg, Spring 1994- Midterm Solution 1 Quality Assurance (40 points) (a) For the failure probabilities given, with general λ , Pr { fail in (0 , T ] } = i T 0 λe - λx dx = 1 - e - λT . It follows that P D = 1 - e - λ b T P FA = 1 - e - λ g T . (b) This is an application of Bayes’ Law. You should have the sketch below in mind: Using the Theorem of Total Probability Pr { rejected } = ( P D ) Pr { bad } + ( P FA ) Pr { good } = (0 . 993)(0 . 01) + (0 . 008)(0 . 99) = 0 . 01785 . Bayes’ Law then yields Pr { bad | rejected } = P D Pr { bad } Pr { rejected } = (0 . 993)(0 . 01) 0 . 01785 = 0 . 5563 . It follows that Pr { good | rejected } = 1 - Pr { bad | rejected } = 0 . 4437 . This may sound bad (i.e. almost half of the computers rejected are good), but when you consider that only a small fraction of the computers are bad it’s not so bad.
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2 K.M. Chugg - October 4, 1995 (c) Using the results of (a), we have P D = 1 - e - λ b T 0 . 99 P FA = 1 - e - λ g T 0 . 01 . Increasing the test time makes it more likely that computers will be rejected; it follows
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Unformatted text preview: that 1-e-λ b T min = 0 . 99 ⇐⇒ T min = 1 λ b ln[1 / (0 . 01)] = 4 . 6 hours 1-e-λ b T max = 0 . 01 ⇐⇒ T max = 1 λ g ln[1 / (0 . 99)] = 6 . 03 hours. (d) Using the result of (c) we have that 1 λ b ln(1 / (0 . 01)) ≤ T ≤ 1 λ g ln(1 / (0 . 99)) , which is only possible when γ = λ b λ g ≥ γ min = ln[1 / (0 . 01)] ln[1 / (0 . 99)] = 458 . 2 . For example, if company’s values are o± by about 15% each so that λ b = 0 . 85 and λ g = 1 / 525, then the test cannot be designed so that the detection and FA requirements are met. 0.5 1 1.5 2 2.5 20 40 60 80 100 EE464: Chugg, Spring ’94 - Midterm 1 Score out of 130 Average = 57.7 0.5 1 1.5 2 2.5 3 3.5 20 40 60 80 100 EE464: Chugg, Spring ’94 - Midterm 1 Score out of 130 Average = 57.7 EE 464 Midterm Solution 3...
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This note was uploaded on 02/02/2010 for the course EE 464 taught by Professor Caire during the Spring '06 term at USC.

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chugg_mt1-94_sol - that 1-e-λ b T min = 0 . 99 ⇐⇒ T...

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