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Statistical independence

Statistical independence. if E 1 and E 2 are s.i. s.i. or. E 1 = flood in 廣東 on June. E 2 = flood in 廣西 on June. E 3 = flood in 哈爾濱 on June. Example:. P(E 1 ) = 0.1; P(E 2 )=0.1; P(E 3 ) = 0.1. E 1 and E 2 are not s.i. E 1 and E 3 are s.i.

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Statistical independence

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  1. Statistical independence if E1 and E2 are s.i. s.i. or

  2. E1 = flood in 廣東 on June E2 = flood in 廣西 on June E3 = flood in 哈爾濱 on June Example: P(E1) = 0.1; P(E2)=0.1; P(E3) = 0.1 E1 and E2 are not s.i. E1 and E3 are s.i.

  3. if E1 and E2 are s.i. if all are s.i.

  4. s.i. and m.e. if E1 and E2 are m.e. if E1 and E2 are s.i.

  5. steel bar steel bar ① ② ring is super ring E2.17 P (failure of this bar system) = ? E1 = bar ① is weak (under strength) E2 = bar ② is weak (under strength) P(failure)

  6. s.i. 0.05 P(E1)=0.05 P(E1|E2)=1 If 5% of bars are weak P(failure) if assume perfectly dependent P(failure)

  7. 1 A B 2 3 C P 2.15 E1 : ①is open P(E1)=2/5 P(E2)=3/4 P(E3)=2/3 P(E3|E2)=4/5 P(E1|E2E3)=1/2 a) P(go from A to B through C)

  8. 1/2 3/5 b) P(go from A to B)

  9. Rainfall magnitude (from hydrologist) small rainfall S medium rainfall M heavy rainfall H P(S) = 2P(M) P(M) = 3P(H) P(S)+P(M)+P(H) = 1 if H P(L) = 0.9 P(L|H) M P(L) = 0.2 P(L|M) S P(L) = 0.05 P(L|S) from geotechnical engineer Theorem of Total Probabilities Example: P (L = landslides in the next storm) = ?

  10. P(L|H) P(H) P(S) P(L|S) P(L|M) P(M) if rainfall magnitude equally likely, P(L) = 1/3(0.9+0.2+0.05)=0.38 P(M) = 0.3 P(S) = 0.6 P(H) = 0.1 P(L) = 0.05×0.6+0.2×0.3+0.9×0.1=0.18

  11. S A En E2 E1 Ei’s are m.e. and c.e. A = AS c.e. = A(E1E2… En) rule = AE1AE2… AEn m.e. P(A) = P(AE1)+P(AE2) +…+P(AEn) = P(A|E1)P(E1)+…

  12. N W E S 70% of traffic will go straight 20% of traffic will go right turn 10% of traffic will go left turn Example: 4 - way stop intersection Given information traffic from E – 60 veh/10min traffic from S - 50 veh/10min traffic from W – 70 veh/10min traffic from N – 20 veh/10min P (next vehicle will go east from the intersection) = ? addition information from similar intersection:

  13. next veh. go east 0 0.2 0.1 0.7 what % of traffic will go east after intersection = 29%

  14. Bayes Theorem • given the result, ask for the likelihood of a specific cause.

  15. Intersection example Suppose a vehicle has just gone east what is the probability that it came from the south? P(S | A) 6.9% of the east bound traffic from the intersection came from the south

  16. Landslide example Suppose landslide occurred, what is the probability that the rain has been just small?

  17. T.O.T (Theorem of Total Probabilities) P(A) = P(A|E1)P(E1)+P(A|E2)P(E2)+…+P(A|En)P(En) Ei’s are m.e. and c.e. Bayes theorem

  18. good enough for construction not a perfect test positive E 2.30 aggregate for construction engineer's judgment based on geology and experience crude test reliability (or quality) is as follows:

  19. After 1 successful test, what is P(G)?

  20. After another successful independent test, P(G)?

  21. 0.7 0.7 0.3 … 5 P(G) After 1 test After 2 tests 1.0000 UST 0.993 0.7 0.95 0.77 0.965 0.9999 HKU 0.3 What if the two tests were performed at the same time?

  22. Supplementary Exercise 2-2-6 (on web page) A landfill containment system A layer of clay and geomembrane to prevent the contaminants leaking into soil stratum • poorly compacted clay layer • holes in geomembrane • extremely heavy rainfall

  23. Leakage will happen: • during extremely heavy rainfall, and either the clay was not well compacted or there were holes in the geomembrane (Event I). • under ordinary rainfalls (i.e. without extremely heavy rainfall), but when the clay was not well compacted and the geomembrane contained holes (Event II).

  24. W = clay well compacted; 90% H = holes in geomembrane; 30% E = extremely heavy rainfall; 20% • The quality of construction has no effect on the future amount of rainfall. E is s.i. of W or H • If the geomembrane contained holes, the probability of a well-compacted clay is reduced to 60%.

  25. I: II: a) Express Event I and II

  26. 0.1 0.3 1-0.6 0.3 b)

  27. c1) W, H m.e.? W, H not m.e. W, H s.i.? W, H not s.i.

  28. I = E W H II = Ē c2) I and II m.e.? Difficult to prove P (I|II) = 0 Method 1. I and II are m.e.

  29. null set Alternatively, I and II are not c.e. because they do not together make up the entire sample space

  30. d) the probability of leakage

  31. down train up train 5:08 5:00 5:18 5:10 5:20 5:28 5:10 5:20 5:00 Puzzle: which friend for dinner?

  32. Example: damage of bridge A small old bridge susceptible to damages from heavy trucks at most two trucks 10% of trucks are overloaded event of overloaded trucks are s.i. O1 : truck 1 is overloaded O2 : truck 2 is overloaded

  33. s O1 O2 a)

  34. b) P(overloaded truck | D) =1 - P(no overloaded truck | D)

  35. c) alt. I – strengthen bridge and also for others P(D) = 0.0064 from 0.0128 alt. II – improve truck inspection overloaded truck from P(D) = 0.0076 which is better?

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