1. Probability and Statistics Modelling Systems of Random Variables – Computer System – Traffic Systems – Financial System – Data System Linear – Weighted sums Non-Linear – Max/min; If-then; products ST2004 2011 Week 101 Forced by randomness, unpredictability

2. Systems Teams in league – Games won by N A N B N C – Sums of binary Student attendance at class – BinaryName Chosen/ Not Chosen – Given chosenPresence/Absence Sets of Dice – Scores X 1 X 2 X 3  Sums, Max S 3 M 3 ST2004 2011 Week 102 Model Decompose Splash Bean Machine

3. Approach 1.Consider simulation thought experiments – Columns and Random Variables – Define/articulate problem Possible Values Transforms, functions Event Identities 2.Define random vars – Algebra for random vars 3.Consider probabilities ST2004 2011 Week 103

4. Sums and Averages: Linear Systems Simple systems – Dice sums, Travel times, Pill Boxes Convergence in simulations – Form running sum; divide by n Estimation by sample survey – Sum; divide by n – Count; divide by n Finance – Accumulation of % changes – Sum, in log scale ST2004 2011 Week 104 Comparisons

5. Max and Min Combinations A system has 3 components A, B and C, with redundancy. It is designed such that it will work if either (C is working) or (both A and B are working). If the lifetimes of A, B and C are 10, 15 and 8 hours, resp, then it will work for 10 hours. ST2004 2011 Week 105

6. Systems of Random Variables Input  Output – Simulation Joint Uncertainty (Input)  Uncertainty(Output) – Prob Dist – Expected Values and Variances ST2004 2011 Week 106 Independent Dependent Linear Non-linear

7. Linear Combs & Normal Distribution Linear Combinations – weighted sums – counts Simple for Normal Normal a useful approx – Central Limit Theorem – SE(mean&prop)  n Convergence 7ST2004 2011 Week 10 Dice sum Dice max

8. Dice: Sums ST2004 2011 Week 108

9. Prob Rules  Prob Dist  ExpVal etc 9ST2004 2011 Week 10

10. 10ST2004 2011 Week 10 Prob Rules  Prob Dist  ExpVal etc

11. cdf for Max Min indep rvs 11ST2004 2011 Week 10

12. Max and Min Combinations A system has 3 components A, B and C, with redundancy. It is designed such that it will work if either (C is working) or (both A and B are working). If the lifetimes of A, B and C are 10, 15 and 8 hours, resp, then it will work for 10 hours. ST2004 2011 Week 1012

13. System Lifetime ST2004 2011 Week 1013 See SystemLife.xlsx

14. Max/min indep random vars cdf(system) via prods based on cdf(comps) – pmf by subtraction, if discrete – pdf by calculus, if continuous Expected Value & Var sumproduct, if discrete calculus, if continuous ST2004 2011 Week 1014

15. Sum/Diff/Lin Comb indep random vars Expected Value & Var simple rules, based on – E[aX+bY]=aE[X] + bE[Y] – Var[aX+bY]=a 2 Var[X] + b 2 E[Y] cdf(system) pmf(system) – by tabulation, enumeration if discrete some special cases – intricate calculus, if continuous some special cases – but often, Normal approx ST2004 2011 Week 1015

16. Theory: Distributions and Lin Combs ST2004 2011 Week 1016

17. 17 Theory: Linear Combinations X,Y random variables a,b constants Z = aX+bY Seek E[Z] and Var[Z] Using Normal (approx) for dist Z? E[Z] and Var[Z] fully specify D iscrete dists only in these notes; extension to continuous dists only a matter of notation; joint pdf instead of joint pmf; integrals instead of sums. ST2004 2011 Week 10

18. 18 Approach via dist Z =Y+X ST2004 2011 Week 10 E[Y]E[X] Var[Y]Var[X] E[Z] Var[Z] In Fill in given Indep

19. 19 Approach via dist Z =Y+X ST2004 2011 Week 10 E[Y] = 2E[X]=4 Var[Y]=2/3Var[X]=4/3 E[Z]= 6Var[Z]=6/3 Cov(X,Y)=0

20. 20 Direct approach: when X,Y indep ST2004 2011 Week 10

21. 21 Approach via dist Z =X+Y ST2004 2011 Week 10 E[Y]E[X] Var[Y]Var[X] Cov[X,Y] E[Z]Var[Z]

22. 22 Approach via dist(Y+X) ST2004 2011 Week 10

23. 23 Direct approach: when X,Y not indep ST2004 2011 Week 10

24. 24ST2004 2011 Week 10 Theory: Expected values for linear combs

25. 25 App: Travel Times ST2004 2011 Week 10

26. Travel Time by Simulation ST2004 2011 Week 1026 T_ABT_BCT_AC 36.824345.529682.3540 31.346139.924571.2705 31.655437.587169.2425 39.418146.030385.4484 37.299249.172186.4713 34.160842.158176.3189 34.364345.295579.6598 36.706946.355683.0624 40.186747.925188.1117 36.722536.041672.7641 34.351743.314977.6666

27. 27 App: Travel Times ST2004 2011 Week 10

28. 28 Travel Times mean354580 var102535 Probs0.1340.1840.088 ST2004 2011 Week 10

29. 29 Times Different? Pr = 0.045 ST2004 2011 Week 10

30. 30 Correlated Travel Times Prob = 0.138 ST2004 2011 Week 10

31. 31 Correlation Prob = 0.023 ST2004 2011 Week 10

32. Proof: discrete case 32ST2004 2011 Week 10

33. Proof: discrete case 33ST2004 2011 Week 10

34. 34 Packing a pillbox ST2004 2011 Week 10

35. Approach by Simulation ST2004 2011 Week 1035

36. Extension 36ST2004 2011 Week 10

37. 37 Pr = 0.32 ST2004 2011 Week 10 Packing a pillbox

38. 38 Common Error ST2004 2011 Week 10

39. Important Special Cases 39ST2004 2011 Week 10

40. Sampling Dists of Avg (S4) ST2004 2011 Week 1040 Convergence at 1/  n Central Limit Theorem Section 5.3,4

41. Simulation Convergence ST2004 2011 Week 1041 Confidence Intervals for Simulations Sec5.7

42. Law of Large Numbers rate of convergence ST2004 2011 Week 1042

43. Theory: Normal Approximation via Central Limit Theorem ST2004 2011 Week 1043

44. 44 Application: sums and averages ST2004 2011 Week 10

45. 45 Application: precision ST2004 2011 Week 10

46. 46 Application: sample size for mean ST2004 2011 Week 10

47. 47 Application: sample size for prop ST2004 2011 Week 10

48. Homework Tijms Q5.2 Someone has written a simulation programme to estimate a probability. 500 runs  estimate of 0.451. If the prob p  0.451, what are SE(est prob)? 95% Conf Int? 1000 runs  estimate of 0.453, what are SE, 95% CI? Is there reason to suspect problem? ST2004 2011 Week 1048

49. Homework Tijms Q5.2 Someone has written a simulation programme to estimate a probability. 500 runs  estimate of 0.451. If the prob = p  0.451, what are EstSE(est prob)? 95% Conf Int? 1000 runs  estimate of 0.453, what are EstSE, 95% CI? Is there reason to suspect problem? No: the difference 0.002 is very small compared to the uncertainties involved ST2004 2011 Week 1049