Chapter 4.pdf

# Normal distribution applications tool failures brake

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Normal Distribution - Applications Tool failures Brake lining wear Tire tread wear Chapter 4 25 It’s the additive effect of temperature variation, material wear, friction, and other random stresses over time, isn’t it?.

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Finding Normal Cumulative Probabilities Chapter 4 26 T z = If T is normally distributed, then let Then z has a normal distribution with a mean of 0 and a standard deviation of 1. The PDF for z is given by 1 2 z - 2 (z) = e 2 Its cumulative distribution is then given by Pr{ } ' z - Z z (z) = (z ) dz =  z is the standardized normal deviate
Normal Probability Tables Z (Z) 1- (Z) -0.55000 0.29116 0.70884 -0.54000 0.29460 0.70540 -0.53000 0.29806 0.70194 -0.52000 0.30153 0.69847 -0.51000 0.30503 0.69497 -0.50000 0.30854 0.69146 -0.49000 0.31207 0.68793 -0.48000 0.31561 0.68439 -0.47000 0.31918 0.68082 -0.46000 0.32276 0.67724 -0.45000 0.32636 0.67364 -0.44000 0.32997 0.67003 Chapter 4 27 Pr{Z < -.5} = .30854 Pr{Z > - .46 = .67724

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Normal Reliability Function Chapter 4 28 2 2 ( ) 2 1 ' 2 t t R(t) = e dt   To find normally distributed reliabilities, we standardize and then go to the normal table. Derive R(t)
Normal Hazard Rate Function Chapter 4 29 f(t) / R(t) IFR Interestingly, the hazard rate function is always increasing. Why? Explain its applications.

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Example Problem - Normal The time to failure of a fan belt is normally distributed with a MTTF = 220 (in hundreds of vehicle miles) and a standard deviation of 40 (in hundreds of vehicle miles). R(100|200) = ?? Median= ?? Chapter 4 30
Student Exercise - Normal The operating hours until failure of a halogen headlamp is normally distributed with a mean of 1200 hr. and a standard deviation of 450 hr. Find: a. The 5 year reliability if normal driving results in the use of the headlamp an average of .2 hr. a day. b. The .90 design life in years. Chapter 4 31

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Time-Dependent Failure Models 3. Lognormal Distribution Chapter 4 32
The Lognormal Failure Process Let T = a random variable, the time to failure. If T has a lognormal distribution, then the logarithm of T has a normal distribution. Chapter 4 33

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Lognormal Density Function Chapter 4 34 ln 0 2 2 MED 1 t - 2 t s 1 f(t) = ; t e 2 s t t med = median time to failure s = shape parameter Define the 2- parameters of the lognormal to be its median (a location parameter) and its shape parameter s.
Lognormal/Normal Relationship Given T is a lognormal random variable, then T Log T Distribution Lognormal Normal Mean ln t med Variance s 2 Mode ln t med Chapter 4 35 2 / med 2 s e t t e e med s s 2 2 2 1 [ ] 2 med mode s t t e =

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