Uncertainty and Safety Measures

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1 Uncertainty and Safety Measures 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 g them How do we represent random variables? Probability distributions and moments

2 Reading assignment Oberkmapf et al. Error and uncertainty in modeling and simulation, Reliability Engineering and System Safety, 75, , 2002 Source: Page11.htm

3 Modeling uncertainty (Oberkampf et al.).

4 . Modeling stages

5 Classification of uncertainties Aleatory uncertainty: t Inherent variability Example: What does regular unleaded cost in Gainesville today? Epistemic uncertainty Lack of knowledge Source: Example: What will be the average cost of regular unleaded January 1, 2012? 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!

6 A slightly different uncertainty t British Airways classification. 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: o te Accident investigations s

8 Many players reduce uncertainty in aircraft structures. t The federal government (e.g. NASA) invests in developing more accurate models and measurement techniques. Boeing invests in higher fidelity simulations and high accuracy manufacturing. Airlines invest in maintenance and inspections. FAA invests in certification of aircraft & pilots. NTSB, FAA and NASA fund accident investigations.

9 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)

10 Histograms and PDF How do you estimate the PDF from a histogram? SOURCE: 611/akerley/question.jpg P robability distribution function f : P( a x a da) f ( x) da

11 Cumulative distribution function x Integral of PDF F( x) P( X x) f( t) dt X = [-3:0.1:3]; p=normcdf(x,0,1) plot(x,p) CDF x

12 Some STATISTICS Mean ( X ) xf ( xdx ) Variance 2 Var ( X ) ( x ) f ( x ) dx Standard deviation Var ( X ) Coefficient of variation COV

13 Uncertainty ypropagationp A function r(x) will have its own PDF

14 problems 1. List at least six safety measures or uncertainty reduction mechanisms used to reduce highway fatalities of automobile drivers. Source: Smithsonian Institution Number: Give examples of aleatory and epistemic uncertainty faced by car designers who want to ensure the safety of drivers. 3. Let x be a standard normal variable N(0,1). 3. Let x be a standard normal variable N(0,1). Calculate the mean and standard deviation of sin(x)

STATISTICS and PROBABILITY

Introduction to Statistics Atatürk University STATISTICS and PROBABILITY LECTURE: PROBABILITY DISTRIBUTIONS Prof. Dr. İrfan KAYMAZ Atatürk University Engineering Faculty Department of Mechanical Engineering