Propagation of Uncertainty

1. Propagation of Uncertainty Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers

2. Introduction • We’ve discussed single-variable probability distributions • This lets us represent uncertain inputs • But what of variables that depend on these inputs? How do we represent their uncertainty? • Some problems can be done analytically; others can only be done numerically • These slides discuss analytical approaches Uncertainty Analysis for Engineers

3. Functions of 1 Random Variable • Suppose we have Y=g(X) where X is a random input variable • Assume the pdf of X is represented by fx. • If this pdf is discrete, then we can just map pdf of X onto Y • In other words X=g-1(Y) • So fy(Y)=fx[g-1(y)] Uncertainty Analysis for Engineers

4. Example • Consider Y=X2. • Also, assume discrete pdf of X is as shown below • When X=1, Y=1; X=2, Y=4; X=3, Y=9 Uncertainty Analysis for Engineers

5. Discrete Variables • Example: • Manufacturer incurs warranty charges for system breakdowns • Charge is C for the first breakdown, C2 for the second failure, and Cx for the xth breakdown (C>1) • Time between failures is exponentially distributed (parameter ), so number of failures in period T is Poisson variate with parameter T • What is distribution for warranty cost for T=1 year Uncertainty Analysis for Engineers

6. Formulation Uncertainty Analysis for Engineers

7. Plots C=2 =1 Uncertainty Analysis for Engineers

8. CDF For Discrete Distributions • If g(x) monotonically increases, then P(Y<y)=P[X<g-1(y)] • If g(x) monotonically decreases, then P(Y<y)=P[X>g-1(y)] • …and, formally, y y x Uncertainty Analysis for Engineers x

9. Another Example • Suppose Y=X2 and X is Poisson with parameter  Uncertainty Analysis for Engineers

10. Continuous Distributions • If fxis continuous, it takes a bit more work Uncertainty Analysis for Engineers

11. Example Normal distribution Mean=0, =1 Uncertainty Analysis for Engineers

12. Example • X is lognormal Normal distribution Uncertainty Analysis for Engineers

13. If g-1(y) is multi-valued… Uncertainty Analysis for Engineers

14. Example (continued) lognormal Uncertainty Analysis for Engineers

15. Example Uncertainty Analysis for Engineers

16. A second example • Suppose we are making strips of sheet metal • If there is a flaw in the sheet, we must discard some material • We want an assessment of how much waste we expect • Assume flaws lie in line segments (of constant length L) making an angle  with the sides of the sheet •  is uniformly distributed from 0 to  Uncertainty Analysis for Engineers

17. Schematic L  w Uncertainty Analysis for Engineers

18. Example (continued) • Whenever a flaw is found, we must cut out a segment of width w Uncertainty Analysis for Engineers

19. Example (continued) • g-1 is multi-valued </2 >/2 Uncertainty Analysis for Engineers

20. Results pdf L=1 cdf Uncertainty Analysis for Engineers

21. Functions of Multiple Random Variables • Z=g(X,Y) • For discrete variables • If we have the sum of random variables • Z=X+Y Uncertainty Analysis for Engineers

22. Example • Z=X+Y Uncertainty Analysis for Engineers

23. Analysis Uncertainty Analysis for Engineers

24. Result Uncertainty Analysis for Engineers

25. Example • Z=X+Y Uncertainty Analysis for Engineers

26. Analysis Uncertainty Analysis for Engineers

27. Compiled Data Uncertainty Analysis for Engineers

28. Example x and y are integers Uncertainty Analysis for Engineers

29. Example (continued) The sum of n independent Poisson processes is Poisson Uncertainty Analysis for Engineers

30. Continuous Variables Uncertainty Analysis for Engineers

31. Continuous Variables Uncertainty Analysis for Engineers

32. Continuous Variables (cont.) Uncertainty Analysis for Engineers

33. Example Uncertainty Analysis for Engineers

34. In General… • If Z=X+Y and X and Y are normal dist. • Then Z is also normal with Uncertainty Analysis for Engineers

35. Products Uncertainty Analysis for Engineers

36. Example • W, F, E are lognormal Uncertainty Analysis for Engineers

37. Central Limit Theorem • The sum of a large number of individual random components, none of which is dominant, tends to the Gaussian distribution (for large n) Uncertainty Analysis for Engineers

38. Generalization • More than two variables… Uncertainty Analysis for Engineers

39. Moments • Suppose Z=g(X1, X2, …,Xn) Uncertainty Analysis for Engineers

40. Moments Uncertainty Analysis for Engineers

41. Moments Uncertainty Analysis for Engineers

42. Approximation Uncertainty Analysis for Engineers

43. Approximation Uncertainty Analysis for Engineers

44. Second Order Approximation Uncertainty Analysis for Engineers

45. Approximation for Multiple Inputs Uncertainty Analysis for Engineers

46. Example • Example 4.13 • Do exact and then use approximation and compare • Waste Treatment Plant – C=cost, W=weight of waste, F=unit cost factor, E=efficiency coefficient Uncertainty Analysis for Engineers

47. Solving… Uncertainty Analysis for Engineers

48. Approximation Uncertainty Analysis for Engineers

49. Variance Uncertainty Analysis for Engineers