L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 4 1 MER301: Engineering Reliability LECTURE 4: Chapter 3: Lognormal Distribution, The Weibull Distribution, and Discrete Variables

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 2 Summary of Topics  Lognormal Distribution  Weibull Distribution  Probability Density and Cumulative Distribution Functions of Discrete Variables  Mean and Variance of Discrete Variables

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 3 Normal Distribution  Many Physical Phenomena are characterized by normally distributed variables  Engineering Examples include variation in such areas as: Dimensions of parts Experimental measurements Power output of turbines

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 4 Lognormal Distribution  Special case of the normal distribution where and the variable w is normally distributed Chemical processes and material properties are often characterized by lognormal distributions  Parameters and are the mean and variance of W, respectively

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 5 Lognormal Distribution

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 6 Lognormal Distribution

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 7 Lognormal Example 4.1  Gas Turbine CO Emissions  is a normally distributed function of combustor fuel/air ratio  Mean value of CO will need to be 9ppm or less

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 8 Lognormal Example 4.1(cont)

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 9 Lognormal Example 4.1  excel spreadsheet for the CO example

L Berkley Davis Copyright Skewness and Kurtosis: Tflame Example  Skewness Skewness characterizes the degree of asymmetry of a distribution around its mean. Positive skewness indicates a distribution with an asymmetric tail extending towards more positive values. Negative skewness indicates a distribution with an asymmetric tail extending towards more negative values" (Microsoft, 1996). Samples from Normal distributions produce a skewness statistic of about zero. ses can be estimated roughly using a formula from Tabachnick & Fidell,1996  Kurtosis kurtosis characterizes the relative peakedness or flatness of a distribution compared to the normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution. Samples from Normal distributions produce a kurtosis statistic of about zero sek can be estimated roughly using a formula from Tabachnick & Fidell, 1996

L Berkley Davis Copyright Lognormal Example 4.1(cont)

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 12 Lognormal Example 4.1(cont)  CO is given by the equation  Let and So that or

L Berkley Davis Copyright Lognormal Example 4.1(cont) MER301: Engineering Reliability Lecture 4 13

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 14 Lognormal Example 4.1(cont)  CO is given by the equation  Test Ln(CO/4.5) and CO for normality ….  Ln(CO/4.5) is normally distributed and CO is not

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 15 Lognormal Example 4.1(cont)  CO is given by the equation  Let and So that or Now we want

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 16 Lognormal Example 4.1(cont)  From the analysis of the data for flame temperature and W  Then

L Berkley Davis Copyright Lognormal Example 4.1(cont)  Summary of the CO Lognormal Distribution MER301: Engineering Reliability Lecture 4 17 Mean CO Standard Deviation of CO System Does not meet CO Requirements -combustor needs a factor of 5 improvement in CO performance

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 18 Weibull Distribution  Widely used to analyze and predict failure for physical systems failure may be a function of time, cycles, starts, landings, etc  Can provide reasonably accurate failure predictions with small samples Important in safety critical systems

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 19 The Weibull Distribution

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 20 The Weibull Failure Function….

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 21 Two parameters define the Weibull distribution: , the shape parameter, is a measure of the time dependency of the probability of failure. Completely random failures(random errors, external shocks) have a  = 1. Failures which increase in probability over time(wearout, old age) have  > 1, and failures whose probability decreases over time(manufacturing errors) have 0 <  < 1., the scale parameter, is the time at which a cumulative 63.2% of the population is expected to have failed The Weibull Distribution

L Berkley Davis Copyright 20009

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 23

L Berkley Davis Copyright Infant Mortality Old Age Useful Life 63.2% Failure Rate at x=delta =1000 X=1000

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 25 Many kinds of failure data plot as a straight line with slope  The x- axis is time and the y-axis is the cumulative failure density function F(t), Weibull plots are used to predict cumulative failures at any time. For instance, with  = 1.66 and = , after time units 5% of the population will have failed. ln(ln(1/(1-63.2%)) = 0. So,  is the y-intercept of the straight line plot. Weibull Plots-Cumulative Density Function t=30000

L Berkley Davis Copyright Weibull Plots-Cumulative Density Function (Two Cycle Weibull Paper) X=100X=1000 X=10000

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 27 Discrete Distribution Probability Mass Function  Describes how the total probability of 1 is distributed among various possible values of the variable X

L Berkley Davis Copyright The Sum of Two Dice…  Define the following probabilities Let A= probability of 2=1/36 Let B= probability of 3=2/36 Let C= probability of 4=3/36 Let D= probability of 5=4/36 Let E= probability of 6=5/36 Let F= probability of 7=6/36 Let H= probability of 9=4/36 Let I= probability of 10=3/36 Let J= probability of 11=2/36 Let K= probability of 12=1/36

L Berkley Davis Copyright For a Probability Mass Function The Sum of Two Dice… The Probability Mass Function  Define the following probabilities Let A= probability of 2=1/36 Let B= probability of 3=2/36 Let C= probability of 4=3/36 Let D= probability of 5=4/36 Let E= probability of 6=5/36 Let F= probability of 7=6/36 Let H= probability of 9=4/36 Let I= probability of 10=3/36 Let J= probability of 11=2/36 Let K= probability of 12=1/36

L Berkley Davis Copyright The Sum of Two Dice… The Cumulative Distribution  Define the following probabilities Let A= probability of 2=1/36 Let B= probability of 3=2/36 Let C= probability of 4=3/36 Let D= probability of 5=4/36 Let E= probability of 6=5/36 Let F= probability of 7=6/36 Let G= probability of 8=5/36 Let H= probability of 9=4/36 Let I= probability of 10=3/36 Let J= probability of 11=2/36 Let K= probability of 12=1/36 MER301: Engineering Reliability Lecture 1 30 i

L Berkley Davis Copyright The Sum of Two Dice… The Mean….  Define the following probabilities Let A= probability of 2=1/36 Let B= probability of 3=2/36 Let C= probability of 4=3/36 Let D= probability of 5=4/36 Let E= probability of 6=5/36 Let F= probability of 7=6/36 Let G= probability of 8=5/36 Let H= probability of 9=4/36 Let I= probability of 10=3/36 Let J= probability of 11=2/36 Let K= probability of 12=1/36 MER301: Engineering Reliability Lecture 1 31

L Berkley Davis Copyright The Sum of Two Dice… The Variance….  Define the following probabilities Let A= probability of 2=1/36 Let B= probability of 3=2/36 Let C= probability of 4=3/36 Let D= probability of 5=4/36 Let E= probability of 6=5/36 Let F= probability of 7=6/36 Let G= probability of 8=5/36 Let H= probability of 9=4/36 Let I= probability of 10=3/36 Let J= probability of 11=2/36 Let K= probability of 12=1/36 MER301: Engineering Reliability Lecture 1 32

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 33 Probability Mass Function 3-29

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 34 Example 4.2  Consider a group of five potential blood donors – A, B, C, D, and E – of whom only A and B have type O + blood. Five blood samples, one from each individual, will be typed in random order until an O + individual is identified. Let X=the number of typings necessary to identify an O + individual Determine the probability mass function of X

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 35 Cumulative Distribution Function 3-31

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 36 Example 4.3  For the previous example (4.2), determine F(x) for each value of x in the set of possible values x=1 to x=4

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 37 Expected Value or Mean of the Discrete Distribution

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 38 Example 4.4  Consider a university having 15,000 students X= number of courses for which a randomly selected student is registered. The probability mass function can be found by knowing how many students signed up for any specific number of classes Determine the probability mass function f(x). Calculate the mean/expected number of courses per student.

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 39 Variance of Discrete Distributions

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 40 Example 4.5  In example 4.4 the density function is given as shown Determine the variance and the standard deviation.

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 41 Expected Value of a Function  If the random variable X has a set of possible values x 1,x 2,…,x n and a probability mass function f(x), the the expected value of a function h(X) can be estimated as where

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 42 Example 4.6  Let X be the number of cylinders in the engine of the next car to be tuned up at a certain facility. The cost of the tune up  h(x)=20+3x+0.5x 2  Assume 50%,30%,and 20% of cars have four, six, and eight cylinders, respectively Since x is a random variable, so is h(x) Write the density function f(y) for y=h(x) Determine the expected value for Y=h(X)

L Berkley Davis Copyright MER301: Engineering Reliability Lecture 4 43 Summary of Topics  Lognormal Distribution  Weibull Distribution  Probability Density and Cumulative Distribution Functions of Discrete Variables  Mean and Variance of Discrete Variables