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Kansrekening en steekproeftheorie Pieter van Gelder TU Delft IVW-Cursus, 16 September 2003.

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Presentatie over: "Kansrekening en steekproeftheorie Pieter van Gelder TU Delft IVW-Cursus, 16 September 2003."— Transcript van de presentatie:

1 Kansrekening en steekproeftheorie Pieter van Gelder TU Delft IVW-Cursus, 16 September 2003

2 De basis van de theorie der kansrekening als fundament voor de cursus; Schatten van verdelingsparameters; Steekproef theorie, waarbij zowel met als zonder voor-informatie wordt gewerkt (Bayesiaanse versus Klassieke steekproeven); Afhankelijkheden tussen variabelen en risico's.

3 Inspection in Civil Engineering

4

5 Stochastic variables

6 Outline What is a stochastic variable? Probability distributions Fast characteristics Distribution types Two stochastic variables Closure

7 Stochastic variable Quantity that cannot be predicted exactly (uncertainty): –Natural variation –Shortage of statistical data –Schematizations Examples: –Strength of concrete –Water level above a tunnel –Lifetime of a chisel –Throw of a dice

8 Relation to events Express uncertainty in terms of probability Probability theory related to events Connect value of variable to event E.g. probability that stochastic variable X –is less than x –is greater than x –is equal to x –is in the interval [x, x+  x] –etc.

9 Probability distribution Probability distribution function = probability P(X  ): F X (  ) = P(X  ) stochast dummy F X ()()  0

10 Probability density Familiar form probability ’distribution’: This is probability density function

11 Probability density Differentiation of F to  : f X (  ) = dF X (  ) / d  f = probability density function f X (  ) d  = P(  < X   +d 

12 F X ()()  fX()fX() P(X    d  P(  < X  

13 F X ()()  fX()fX() P(X   

14 Discrete and continuous discrete variable: p X (x) x x F X (x) continuous variable: x f X (x) F X (x) x probability density (cumulative) probability distribution

15 Fast characteristics x f X (x) XX XX  X mean, indication of location  X standard deviation, indication for spread

16 Fast characteristics x XX XX Mean  location maximum (mode) f X (x)

17 Fast characteristics Mean (centre of gravity) Variance Standard deviation Coefficient of variation

18 Normal distribution x f X (x) Normal distributions XX XX XX Completely determined by mean and standard deviation

19 Normal distribution Probability density function Standard normally distributed variable (often denoted by u):

20 Normal distribution Why so popular? Central limit theorem: Sum of many variables with arbitrary distributions is (almost) normally distributed. Convenient in structural reliability calculations

21 Two stochastic variables joint probability density function

22 Contour map probability density x y

23 Two stochastic variables Relation to events dd dd

24 Example Health survey. Measurements of: Length Weight lengte (m) kansdichtheid (1/m) gewicht (kg) kansdichtheid (1/kg)

25 Logical contour map? length (m) weight (kg)

26 Dependency length (m) weight (kg)

27 Fast characteristics Location:  X,  Y means Spread  X,  Y standard deviation Dependency cov XY covariance  XY = cov XY /  X  Y correlation, between -1 and 1

28 Independent variables

29 Closure of the short Introduction to Stochastics What is a stochastic variable? Probability distributions Fast characteristics Distribution types Two stochastic variables

30 Parameter estimation methods Given a dataset x 1, x 2, …, x n Given a distribution type F(x|A,B,…) How to estimate the unknown parameters A,B,… to the data?

31 List of estimation methods MoM ML LS Bayes

32 MoM Distribution moments = Sample moments  x n f(x)dx =  x i n F(x) = 1- exp[-(x-A)/B] A MOM = std(x) B MOM = mean(x) +std(x)

33 Binomial distribution X~Bin(N,p) The binomial distribution gives the discrete probability distribution of obtaining exactly n successes out of N Bernoulli trials (where the result of each Bernoulli trial is true with probability p and false with probability q=1-p). The binomial distribution is therefore given bydiscrete probability distributionBernoulli trialsBernoulli trial f X (n) =

34 E(X) = Np; var(X)=Npq

35 MoM-estimator of p p MOM =  x i / N for j=1:M, X=0; for I=1:N, if rand(1)

36 Case Study Webtraffic statistics –The number of pageviews on websites

37 Statistics on Usage of Screen sizes Is it necessary to download from every user his/her screen size? Is it sufficient to inspect the screen size of just N users, and still have a reliable percentage of the used screen sizes?

38 Assume 41% of the complete population uses size 1024x768 Inspection population size N = 100, 1000, …and simulate the results by generating the usage from a Binomial distribution. Theoretical analysis: Cov=sqrt(1/p - 1)N -1/2

39 Coefficient of variations (as a function of p and N) PNPN %11.75%3.7%1.2%0.1% 39.8%12.3%3.9%1.3%0.1% 6.2%38.9%12.3%3.9%0.4% 5.4%41.8%13.2%4.2%0.4% 3.2%55.0%17.4%5.5%0.55%

40 Optimisation of the inspection sample size Assume the costs of getting screen size information from a user is A Assume the costs of having a larger cov-value is B TC(N) = A.N + B.sqrt(1/p - 1)N -1/2 The optimal sample size follows from TC’(N) = 0, giving N* = B/2A.(1/p - 1) -2/3 For this choice of N, the cov = (2A/B.(1/p – 1)) 1/3

41 Case study container inspectie Toelaatbare ‘ontglip kans’ p = 1/1.000 containers Populatie bestaat uit containers Inspectie bestaat uit controle van containers Stel dat 1 container uit deze steekproef wordt afgekeurd Dan is p MOM =0.001, maar std(p MoM )= Als std(p MoM )<0.001, dan inspectie van volledige populatie (immers std(p MoM )=sqrt(pq)  sqrt(1/N))

42 Inspectie volledige populatie (bij kleine p-waarden) Inspectiekosten moeten zich terugverdienen uit de boete-opbrengsten Inspectiekosten: x K/C Opbrengst zonder inspectie: NI (Negative Impact) Opbrengst met inspectie: p x x boete – x K/C p x x boete – x K/C > NI

43 Bayesian statistics P(A|B)=P(A and B)/P(B) P(A|B)=P(B|A)P(A)/P(B) A = parameters B = data


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