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 ST Week 101 Forced by randomness, unpredictability

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 ST Week 102 Model Decompose Splash Bean Machine

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 ST Week 103

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 ST Week 104 Comparisons

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. ST Week 105

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

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

Dice: Sums ST Week 108

Prob Rules  Prob Dist  ExpVal etc 9ST Week 10

10ST Week 10 Prob Rules  Prob Dist  ExpVal etc

cdf for Max Min indep rvs 11ST Week 10

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. ST Week 1012

System Lifetime ST Week 1013 See SystemLife.xlsx

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 ST Week 1014

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 ST Week 1015

Theory: Distributions and Lin Combs ST Week 1016

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. ST Week 10

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

19 Approach via dist Z =Y+X ST 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 Direct approach: when X,Y indep ST Week 10

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

22 Approach via dist(Y+X) ST Week 10

23 Direct approach: when X,Y not indep ST Week 10

24ST Week 10 Theory: Expected values for linear combs

25 App: Travel Times ST Week 10

Travel Time by Simulation ST Week 1026 T_ABT_BCT_AC

27 App: Travel Times ST Week 10

28 Travel Times mean var Probs ST Week 10

29 Times Different? Pr = ST Week 10

30 Correlated Travel Times Prob = ST Week 10

31 Correlation Prob = ST Week 10

Proof: discrete case 32ST Week 10

Proof: discrete case 33ST Week 10

34 Packing a pillbox ST Week 10

Approach by Simulation ST Week 1035

Extension 36ST Week 10

37 Pr = 0.32 ST Week 10 Packing a pillbox

38 Common Error ST Week 10

Important Special Cases 39ST Week 10

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

Simulation Convergence ST Week 1041 Confidence Intervals for Simulations Sec5.7

Law of Large Numbers rate of convergence ST Week 1042

Theory: Normal Approximation via Central Limit Theorem ST Week 1043

44 Application: sums and averages ST Week 10

45 Application: precision ST Week 10

46 Application: sample size for mean ST Week 10

47 Application: sample size for prop ST Week 10

Homework Tijms Q5.2 Someone has written a simulation programme to estimate a probability. 500 runs  estimate of 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? ST Week 1048

Homework Tijms Q5.2 Someone has written a simulation programme to estimate a probability. 500 runs  estimate of 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 is very small compared to the uncertainties involved ST Week 1049