1. RELIABILITY ENGINEERING IN MECHANICAL ENGINEERING 3 조 20100185 구태균 20111027 김원식 PROJECT 1

2. Objective Select the best distribution of given data

3. C.D.F. estimation method Types of C.D.F Symmetric simple cumulative distributionF(x j ) = ( j - 0.5) / n Mean rankF(x j ) = j /(n +1) Median rankF(x j ) = ( j - 0.3) /(n + 0.4) The rest methodF(x j ) = ( j - 0.375) /(n + 0.25)

4. Probability distribution Types of Probability Distribution Functions Normal distribution LogNormal distribution Weibull distribution Bi-exponential distribution

5. DATA SET 1 N = 12 115, 156, 272, 313, 338, 350, 442, 528, 540, 585, 672, 48

6. Symmetric simple cumulative distribution Normal Bi-exponential LogNormal Weibull Data set 1

7. Mean rank Normal Bi-exponential LogNormal Weibull Data set 1

8. Median rank Normal Bi-exponential LogNormal Weibull Data set 1

9. The rest method Normal Bi-exponential LogNormal Weibull Data set 1

10. Comparing the R 2 values R2R2 SymmetricMeanMedianOthers Normal0.974640.981680.978640.97742 LogNormal0.881530.855820.859170.86006 Weibull0.961440.942830.953520.95644 Bi-exponential0.939620.962580.952010.94809 Normal, Weibull, and Bi-exponential distributions look promising Data set 1

11. D n α Values N=12 αD n α (Normal)D n α (Weibull, Bi-exponential) 0.050.2420.247 0.150.2100.213 Data set 1

12. Symmetric simple cumulative distribution Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 363.25 ▷ σ = 188.1294 LogNormal ▷ μ = 5.686068 ▷ σ = 0.745131 Weibull ▷ m = 1.54908 ▷ ξ = 421.5788 Bi-exponential ▷ ξ = 164.7446 ▷ x 0 = 454.7216 Data set 1 Normal Bi-exponential LogNormal Weibull

13. Mean rank Data set 1 Normal Bi-exponential LogNormal Weibull Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 363.25 ▷ σ = 188.1294 LogNormal ▷ μ = 5.686068 ▷ σ = 0.745131 Weibull ▷ m = 1.28485 ▷ ξ = 436.126 Bi-exponential ▷ ξ = 194.5525 ▷ x 0 = 460.9883

14. Median rank Data set 1 Normal Bi-exponential LogNormal Weibull Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 363.25 ▷ σ = 188.1294 LogNormal ▷ μ = 5.686068 ▷ σ = 0.745131 Weibull ▷ m = 1.42327 ▷ ξ = 427.6218 Bi-exponential ▷ ξ = 177.6199 ▷ x 0 = 457.5311

15. The rest method Data set 1 Normal Bi-exponential LogNormal Weibull Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 363.25 ▷ σ = 188.1294 LogNormal ▷ μ = 5.686068 ▷ σ = 0.745131 Weibull ▷ m = 1.46618 ▷ ξ = 425.4025 Bi-exponential ▷ ξ = 173.0104 ▷ x 0 = 456.5952

16. DATA SET 2 N = 10 472, 33, 647, 306, 220, 422, 23, 413, 533, 309

17. Symmetric simple cumulative distribution Normal Bi-exponential LogNormal Weibull Data set 2

18. Mean rank Normal Bi-exponential LogNormal Weibull Data set 2

19. Median rank Normal Bi-exponential LogNormal Weibull Data set 2

20. The rest method Normal Bi-exponential LogNormal Weibull Data set 2

21. Comparing the R 2 values R2R2 SymmetricMeanMedianOthers Normal0.954230.957520.956470.95586 LogNormal0.729720.726230.728550.72906 Weibull0.842650.824630.835320.83811 Bi-exponential0.939920.957110.949780.94678 Normal and Bi-exponential distributions look promising Data set 2

22. D n α Values N=10 αD n α (Normal)D n α (Weibull, Bi-exponential) 0.050.2620.265 0.150.2280.230 Data set 2

23. Symmetric simple cumulative distribution Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 337.8 ▷ σ = 203.36 LogNormal ▷ μ = 5.45 ▷ σ = 1.17 Weibull ▷ m = 0.97002 ▷ ξ = 408.290 Bi-exponential ▷ ξ = 171.233 ▷ x 0 = 431.839 Data set 2 Normal Bi-exponential LogNormal Weibull

24. Mean rank Data set 2 Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 337.8 ▷ σ = 203.36 LogNormal ▷ μ = 5.45 ▷ σ = 1.17 Weibull ▷ m = 0.78843 ▷ ξ = 434.41 Bi-exponential ▷ ξ = 207.04 ▷ x 0 = 440.161 Normal Bi-exponential LogNormal Weibull

25. Median rank Data set 2 Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 337.8 ▷ σ = 203.36 LogNormal ▷ μ = 5.45 ▷ σ = 1.17 Weibull ▷ m = 0.88319 ▷ ξ = 419.120 Bi-exponential ▷ ξ = 181.159 ▷ x 0 = 434.391 Normal Bi-exponential LogNormal Weibull

26. The rest method Data set 2 Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 337.8 ▷ σ = 203.36 LogNormal ▷ μ = 5.45 ▷ σ = 1.17 Weibull ▷ m = 0.91274 ▷ ξ = 415.277 Bi-exponential ▷ ξ = 181.159 ▷ x 0 = 434.391 Normal Bi-exponential LogNormal Weibull

27. DATA SET 1+2 N = 22 115, 156, 272, 313, 338, 350, 442, 528, 540, 585, 672, 48, 472, 33, 647, 306, 220, 422, 23, 413, 533, 309

28. Symmetric simple cumulative distribution Normal Bi-exponential LogNormal Weibull Data set 1+2

29. Mean rank Normal Bi-exponential LogNormal Weibull Data set 1+2

30. Median rank Normal Bi-exponential LogNormal Weibull Data set 1+2

31. The rest method Normal Bi-exponential LogNormal Weibull Data set 1+2

32. Comparing the R 2 values R2R2 SymmetricMeanMedianOthers Normal0.969820.978860.974760.97321 LogNormal0.796270.793110.795280.79573 Weibull0.915830.904020.911550.9133 Bi-exponential0.934680.960230.948190.94386 Normal, Weibull, and Bi-exponential distributions look promising Data set 1+2

33. D n α Values N=22 αD n α (Normal)D n α (Weibull, Bi-exponential) 0.050.18440.1894 0.150.16020.1636 Data set 1+2

34. Symmetric simple cumulative distribution Data set 1+2 Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 351.6818 ▷ σ = 190.7434 LogNormal μ = 5.576894285 σ = 0.934754244 Weibull ▷ m = 1.24946 ▷ ξ = 415.1585 Bi-exponential ▷ ξ = 161.8123 ▷ x 0 = 443.1602 Normal Bi-exponential LogNormal Weibull

35. Mean rank Data set 1+2 Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 351.6818 ▷ σ = 190.7434 LogNormal μ = 5.576894285 σ = 0.934754244 Weibull ▷ m = 1.09669 ▷ ξ = 427.2008 Bi-exponential ▷ ξ = 180.8318 ▷ x 0 = 446.9711 Normal Bi-exponential LogNormal Weibull

36. Median rank Data set 1+2 Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 351.6818 ▷ σ = 190.7434 LogNormal μ = 5.576894285 σ = 0.934754244 Weibull ▷ m = 1.17844 ▷ ξ = 420.229 Bi-exponential ▷ ξ = 170.068 ▷ x 0 = 444.8997 Normal Bi-exponential LogNormal Weibull

37. The rest method Data set 1+2 Bold: Theoretical curve Straight: α=0.15 Dotted: α=0.05 Normal ▷ μ = 351.6818 ▷ σ = 190.7434 LogNormal μ = 5.576894285 σ = 0.934754244 Weibull ▷ m = 1.203 ▷ ξ = 418.3809 Bi-exponential ▷ ξ = 166.9449 ▷ x 0 = 443.8614 Normal Bi-exponential LogNormal Weibull

38. Conclusions According to the R 2 values, normal distribution looks most promising for all sets of data. This is confirmed by the K-S test, all three data sets for normal distribution safely falling with in α=0.15 significance level.

39. Thank You