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

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 January 1, 2014? 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: / /

A slightly different uncertainty classification. British Airways Distinction between Acknowledged and Unacknowledged errors Type of uncertainty DefinitionCausesReduction measures ErrorDeparture of average from model Simulation errors, construction errors Testing and model refinement VariabilityDeparture of individual sample from average Variability in material properties, construction tolerances Tighter tolerances, quality control

Modeling and Simulation.

Error modeling

Uncertainty reduction measures Design: Refined simulation models, building block tests. Aleatory or epistemic? Manufacture: Quality control. A or E? Operation: Licensing of operators, maintenance and inspections. A or E? Post-mortem: Accident investigations. A or E? Living with uncertainties by using safety factors

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)

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.

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.

Number of boxes

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

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

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

Moments Mean Variance Standard deviation Coefficient of variation Skewness

Questions 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?