2. How do we classify uncertainties? What are their sources? – Lack of knowledge vs. variability. What type of safety measures do we take? – Design, manufacturing, operations & post- mortems – Living with uncertainties vs. changing them How do we represent random variables? – Probability distributions and moments Uncertainty and Safety Measures

3. Reading assignment Oberkmapf et al. “Error and uncertainty in modeling and simulation”, Reliability Engineering and System Safety, 75, 333-357, 2002 S-K Choi, RV Grandhi, and RA Canfield, Reliability-based structural design, Springer 2007. Chapter 1 and Section 2.1. Available on-line from UF library http://www.springerlink.com/content/w62672/#section=3200 07&page=1 http://www.springerlink.com/content/w62672/#section=3200 07&page=1 Source: www.library.veryhelpful.co.uk/ Page11.htm

4. Modeling uncertainty (Oberkampf et al.).

5. Classification of uncertainties Aleatory uncertainty: Inherent variability –Example: What does regular unleaded cost in Gainesville today? Epistemic uncertainty Lack of knowledge –Example: What will be the average cost of regular unleaded next January 1? Distinction is not absolute Knowledge often reduces variability –Example: Gas station A averages 5 cents more than city average while Gas station B – 2 cents less. Scatter reduced when measured from station average! Source: http://www.ucan.org/News/UnionTrib /http://www.ucan.org/News/UnionTrib /

6. A slightly different uncertainty classification. British Airways 737-400 Distinction between Acknowledged and Unacknowledged errors

7. Safety measures Design: Conservative loads and material properties, accurate models, certification of design Manufacture: Quality control, oversight by regulatory agency Operation: Licensing of operators, maintenance and inspections Post-mortem: Accident investigations

8. Airlines invest in maintenance and inspections. A or E? FAA certifies aircraft & pilots. A or E? NTSB, FAA and NASA fund accident investigations. Boeing performs higher fidelity simulations and high accuracy manufacturing. A or E? The federal government (e.g. NASA) develops more accurate models and measurement techniques. A or E? Many players invest to reduce uncertainty in aircraft structures.

9. problems uncertainty 1.List at least six safety measures or uncertainty reduction mechanisms used to reduce highway fatalities of automobile drivers. 2.Give examples of aleatory and epistemic uncertainty faced by car designers who want to ensure the safety of drivers. Source: Smithsonian Institution Number: 2004-57325

10. Representation of uncertainty Random variables: Variables that can take multiple values with probability assigned to each value Representation of random variables –Probability distribution function (PDF) –Cumulative distribution function (CDF) –Moments: Mean, variance, standard deviation, coefficient of variance (COV)

11. Probability density function (PDF) If the variable is discrete, the probabilities of each value is the probability mass function. For example, with a single die, toss, the probability of getting 6 is 1/6.If you toss a pair of dice the probability of getting twelve (two sixes) is 1/36, while the probability of getting 3 is 1/18. The PDF is for continuous variables. Its integral over a range is the probability of being in that range.

12. Histograms Probability density functions have to be inferred from finite samples. First step is a histogram. Histograms divide samples to finite number of ranges and show how many samples in each range (box) Histograms below generated from normal distribution with 50 and 500,000 samples.

13. Number of boxes

14. Histograms and PDF How do you estimate the PDF from a histogram? Only need to scale.

15. Cumulative distribution function Integral of PDF Experimental CDF from 500 samples shown in blue, compares well to exact CDF for normal distribution.

16. Problems CDF Our random variable is the number seen when we roll one die. What is the CDF of 2? Our random variable is the sum on a pair of dice. What is the CDF of 2? Of 13?

17. Probability plot A more powerful way to compare data to a possible CDF is via a probability plot (500 points here)

18. Moments Mean Variance Standard deviation Coefficient of variation Skewness