a. Explanation Calculation: The lifetime of a semiconductor laser follows a lognormal distribution with mean 7,000 and standard deviation 600. Lognormal distribution: If W follows normal distribution with mean θ and the variance ω 2 , then X = exp ( W ) follows lognormal distribution with probability density function . f ( x ) = 1 x ω 2 π e x p [ − ( ln ( x ) − θ ) 2 ω 2 ] , 0 < x < ∞ with mean E ( X ) = exp ( θ + ω 2 2 ) and variance V ( X ) = exp ( 2 θ + ω 2 ) ( exp ( ω 2 ) − 1 ) . Hence, E ( X ) = exp ( θ + ω 2 2 ) = 7 , 000 and V ( X ) = exp ( 2 θ + ω 2 ) ( exp ( ω 2 ) − 1 ) = 360 , 000 . For solving θ and ω 2 , Let x = exp ( θ ) and y = exp ( w 2 ) . Hence, exp ( θ + ω 2 2 ) = 7 , 000 exp ( θ ) exp ( ω 2 2 ) = 7 , 000 exp ( θ ) exp ( ω 2 ) = 7 , 000 x y = 7 , 000 (1) Now, exp ( 2 θ + ω 2 ) ( exp ( ω 2 ) − 1 ) = ( 600 ) 2 ( exp b. To determine Find the probability that the lifetime in hours that 90% of the lasers exceed.

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118539712

Author: Douglas C. Montgomery

Publisher: WILEY

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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 4, Problem 216SE

To determine

Find the probability that a laser fails before 5,800 hours.

To determine

Find the probability that the lifetime in hours that 90% of the lasers exceed.

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Chapter 4, Problem 215SE

Chapter 4, Problem 217SE

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