1 Lecture 13: Other Distributions: Weibull, Lognormal, Beta; Probability Plots Devore, Ch. 4.5 – 4.6

Topics I.Weibull Distribution II.Lognormal Distribution III.Beta Distribution IV.Probability Plots

I. Weibull Model Usage Weibull is a good general purpose distribution -- it may be used to represent a variety of distributions. –Used mostly because of its ability to fit data rather than its theoretical justification. –Very common tool in reliability prediction CDF  1 - e -(x/  )  –  scale parameter –  shape parameter shape parameter as it relates to reliability prediction.  <1; decreasing failure rate with usage  =1; constant failure rate – EXPONENTIAL!  >1; increasing failure rate 3 <  < 4 ; approaches symmetrical - normal (  =3.5)

Weibull Distribution Shapes Turns out the Exponential Distribution is also a special case of the Weibull. Weibull - based on a shape parameter~  ; Scale parameter ~  Weibull for Different Shape Parameters time prob density function, pdf  =.5  =1  =2  =3.5  = 0.5  = 1  = 3.5

Weibull Properties PDF CDF E(X) V(X)

Weibull Example Let T represent time to failure (hours) of a bearing in a mechanical shaft. T may be modeled using Weibull with  = 0.5,  5000 hrs/failures. –Find E(T) –Determine the probability that a bearing will last at least 6000 hours (reliability: R(t) = P(T > t)).

Mean-time-to-failure (MTTF) In reliability, the expected value of a variable following a Weibull (or any other failure distribution, such as the exponential) is often referred to as the mean time to failure (MTTF). For example, if T is the time to failure for a certain component and it follows a Weibull distribution with  =1, what is the probability that a component will last more than the MTTF?

Practical Uses of Weibull Also used to model probabilities of building excess (excess material, excess capacity). Let X be the value that represents excess. Example: suppose you have material thickness requirement of 7 mm. From historical data you fit X to a Weibull with parameters  = 2,  4. What is the likelihood of producing > 10 mm? (3 in excess of the 7mm)

II. Lognormal Distribution A nonnegative rv is lognormal if the rv Y = ln(X) has a normal distribution so that ln(X)~N( , σ ) PDF CDF

Expected Value and Variance of the Lognormal E(X) V(X) Note:  and σ are the mean and standard deviation of Y or ln(X), not of X!

Lognormal - Key Characteristics X may not be normal but ln(x) needs to be normal. Lognormal curves - positive skew

Lognormal Example Suppose a mechanical component has a wear-out effect that follows a lognormal distribution where  =8.5;  = 0.20 What is the probability that the unit will last at least 3000 hours?

III. The “Beta” Distribution Similar to uniform, if X follows a beta distribution, is defined in a finite interval [A,B] PDF: The case of A=0 and B=1, gives the standard beta distribution

Expected Value and Variance of the Beta Distribution E(X): V(X):

Common Applications of the Beta PDF: Project Management –A --> optimistic completion time for a task –B --> pessimistic completion time for task Common Use of Standard Beta –A = 0 and B=1 –X is proportion between 0 and 1. –Examples: % of time, % mixture.

IV. Probability Plots Used to check a distribution assumption –In real life, it is hard to figure out a distribution, typically we need sample data to check and fit a distribution If a distribution actually follows the assumed distribution, the points in the plot will follow a straight line (45°)

How to? By Hand: –Order n observations from smallest to largest –Plot the following pairs: ([100(i - 0.5)/n] th percentile, i th smallest observation in sample) For i = 1.. n –Example: suppose you have 10 observations. 1st sample value Vs. [100(1-0.5)/10]=5th percentile (1 st obs vs. 5 th percentile) Plot pairs for each actual value in sample. If all points fall on 45 o line, then distribution under choice is perfect fit. The further points are scattered about line, the less likely the data may be fit by the selected distribution (used to generate the percentile values) But mostly with software (Minitab!)

Chapter 4; Problem 90 Data Vs. NormalData Vs. Exponential

Analyzing Probability Plots Look for straight lines. One challenge - lots of distributions may fit data. May want to search for best fit. Use Minitab to compare distribution fits with probability plots. (in Minitab, select lowest AD score) –Use data from problem 90 and compare fits of Normal, Exponential, Weibull, Lognormal.

Solutions Slide 6-Weibull –E(X) = 5000  k –P(X > 6000) = 1 – P(X <= 6000) –W(6000; 5000,0.5) Exp(-6000/5000)^.5 = Lognormal –Z = (ln(3000) / 0.2) = 1 - (z=-2.55 ) =.99461