1. Propagation of Uncertainty Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers1

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 Engineers2

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 f x. If this pdf is discrete, then we can just map pdf of X onto Y In other words X=g -1 (Y) So f y (Y)=f x [g -1 (y)] Uncertainty Analysis for Engineers3

4. Example Consider Y=X 2. 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 Engineers4

5. Discrete Variables Example: ◦ Manufacturer incurs warranty charges for system breakdowns ◦ Charge is C for the first breakdown, C 2 for the second failure, and C x for the x th 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 Engineers5

6. Formulation 6

7. Plots 7 C=2 =1

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 g -1 (y)] …and, formally, Uncertainty Analysis for Engineers8 x y x y

9. Another Example Suppose Y=X 2 and X is Poisson with parameter Uncertainty Analysis for Engineers9

10. Continuous Distributions If f x is continuous, it takes a bit more work Uncertainty Analysis for Engineers10

11. Example Uncertainty Analysis for Engineers11 Normal distribution Mean=0,  =1

12. Example X is lognormal Uncertainty Analysis for Engineers12 Normal distribution

13. If g -1 (y) is multi-valued… Uncertainty Analysis for Engineers13

14. Example (continued) Uncertainty Analysis for Engineers14 lognormal

15. Example Uncertainty Analysis for Engineers15

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 Engineers16

17. Schematic Uncertainty Analysis for Engineers17 L  w

18. Example (continued) Whenever a flaw is found, we must cut out a segment of width w Uncertainty Analysis for Engineers18

19. Example (continued) g -1 is multi-valued Uncertainty Analysis for Engineers19  <  /2  >  /2

20. Results Uncertainty Analysis for Engineers20 L=1 cdf pdf

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 Engineers21

22. Example Z=X+Y Uncertainty Analysis for Engineers22

23. Analysis XYZPZ-rank 11011.081 12021.044 13031.087 21012.242 22022.125 23032.248 31013.083 32023.046 33033.089 Uncertainty Analysis for Engineers23

24. Result Uncertainty Analysis for Engineers24

25. Example Z=X+Y Uncertainty Analysis for Engineers25

26. Analysis XYZPZ-rank 123.081 134.042 145.083 224.242 235.123 246.244 325.083 336.044 347.085 Uncertainty Analysis for Engineers26

27. Compiled Data zfz 3.08 4.28 5 6 7.08 Uncertainty Analysis for Engineers27

28. Example Uncertainty Analysis for Engineers28 x and y are integers

29. Example (continued) Uncertainty Analysis for Engineers29 The sum of n independent Poisson processes is Poisson

30. Continuous Variables Uncertainty Analysis for Engineers30

31. Continuous Variables Uncertainty Analysis for Engineers31

32. Continuous Variables (cont.) Uncertainty Analysis for Engineers32

33. Example Uncertainty Analysis for Engineers33

34. In General… If Z=X+Y and X and Y are normal dist. Then Z is also normal with Uncertainty Analysis for Engineers34

35. Products Uncertainty Analysis for Engineers35

36. Example W, F, E are lognormal Uncertainty Analysis for Engineers36

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 Engineers37

38. Generalization More than two variables… Uncertainty Analysis for Engineers38

39. Moments Suppose Z=g(X 1, X 2, …,X n ) Uncertainty Analysis for Engineers39

40. Moments Uncertainty Analysis for Engineers40

41. Moments Uncertainty Analysis for Engineers41

42. Approximation Uncertainty Analysis for Engineers42

43. Approximation Uncertainty Analysis for Engineers43

44. Second Order Approximation Uncertainty Analysis for Engineers44

45. Approximation for Multiple Inputs Uncertainty Analysis for Engineers45

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 Engineers46 mediancov W2000 ton/y.2 F$20/ton.15 E1.6.125

47. Solving… Uncertainty Analysis for Engineers47

48. Approximation Uncertainty Analysis for Engineers48

49. Variance Uncertainty Analysis for Engineers49