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1、Elements of Probability TheoryDr. Dagang LuProfessorSchool of Civil EngineeringHarbin Institute of Technology1 Random Variables 1 Random Variables 1 11.1 Definition of Random VariablesLet be a random experiment, is its sample space, for , there exists a real single-value function .If for ,is a rando
2、m event on event field,then we can call a random variable.random variablediscrete random variablecontinuous random variableor1.2 Description of Random Variables1. Probability Mass Function (PMF)for discrete random variableThere are three kinds of functions for description of random variables as foll
3、ows:2. Cumulative Distribution Function (CDF)for discrete random variablefor continuous random variable3. Probability Density Function (PDF)for continuous random variable1 Random Variables 1 Random Variables 2 21 Random Variables 1 Random Variables 3 31. Probability Mass Function (PMF)The probabilit
4、y mass function (PMF) is defined for discrete random variables as follows: represents probability that a discrete random variable is equal to a specific value , where is a real number. Mathematically,2. Cumulative Distribution Function (CDF) represents the total sum (or integral) of all probability
5、functions (continuous and discrete) corresponding to values less than or equal to . Mathematically,The cumulative distribution function (CDF) is defined for both discrete and continuous random variables as follows:1 Random Variables 1 Random Variables 4 43. Probability Density Function (PDF)For cont
6、inuous random variables, the probability density function (PDF) is defined as the first derivative of the cumulative function . Mathematically,Properties of CDF, PDF and PMF1.The CDF is a positive, nondecreasing function whose value is between 0 and 1:2.If , then3. , 4.For continuous random variable
7、, 1 Random Variables 1 Random Variables 5 51.3 Moments of Random Variables1. Mean or Expected Value (First Moment)The mean value of a random variable is denoted byFor a continuous random variable, the mean value is defined asFor a discrete random variable, the mean value is defined asThe expected va
8、lue of is commonly denoted by and is equal to the mean value of1 Random Variables 1 Random Variables 6 6For a discrete random variable, the nth moment is defined asFor a continuous random variable, the nth moment is defined asThe expected value of is called the nth moment ofThe mathematical expectat
9、ion of an arbitrary function of the random variable is defined as1 Random Variables 1 Random Variables 7 7The standard deviation of is defined as the positive square root of the variance:An important formulaThe variance of a random variable is a measure of the degree of randomness about the mean val
10、ue:2. Variance and Standard Deviation (Second Moment)continuous variablediscrete variable1 Random Variables 1 Random Variables 8 8If a set of n observations are obtained for a particular random variable , then The non-dimensional coefficient of deviation is defined as the standard deviation divided
11、by the mean:3. Moments of Samplethe true mean can be approximated by the sample meanthe true standard deviation can be approximated by the sample standard deviation1 Random Variables 1 Random Variables 9 91.4 Standard Form of Random VariablesLet be a random variable. The standard form of , denoted b
12、y , is defined asthe mean value of is calculated as follows:the deviation of is calculated as follows:2 Common Probability Models2 Common Probability Models112.1 Uniform Random Variables2 Common Probability Models2 Common Probability Models222.2 Normal Random VariablesFor a standard normal RV UFor a
13、 general normal RV X The PDF of U is denoted by The CDF of U is denoted by 2 Common Probability Models2 Common Probability Models33Standard Normal Random VariablesRelationship between general normal RV X and standard normal RV U2 Common Probability Models2 Common Probability Models44Properties of di
14、stribution function of a normal RV The PDF is symmetrical about the mean value The sum of and is equal to 1The inverse CDF of a normal RV2 Common Probability Models2 Common Probability Models552.3 Lognormal Random VariablesCDF &PDF lognormal RVsDefinition of Lognormal RVs If is normally distributed
15、, then is a lognormal RV ( ) .2 Common Probability Models2 Common Probability Models66Moments of Lognormal RVs If , thenProperties of Lognormal RVs2 Common Probability Models2 Common Probability Models772.4 Gamma DistributionPDF of Gamma RVsGamma Function for Moments of Gamma RVs , are distribution
16、parameters2 Common Probability Models2 Common Probability Models882.5 Extreme Type (Gumbel Distribution)CDF & PDF of Extreme RVs for Moments of Extreme RVs , are distribution parameters2 Common Probability Models2 Common Probability Models992.6 Extreme Type CDF & PDF of Extreme RVs for Moments of Ex
17、treme RVs , are distribution parameters for for 2 Common Probability Models2 Common Probability Models10102.7 Extreme Type (Weibull Distribution)CDF of the Largest Values for Moments of the Largest Values , , are distribution parameters2 Common Probability Models2 Common Probability Models11112.7 Ex
18、treme Type (Weibull Distribution) CDF of the Smallest Values for Moments of the Smallest Values , , are distribution parameters2 Common Probability Models2 Common Probability Models12122.8 Poisson DistributionProperties of Poisson DistributionAssumptions of Poisson Distribution It is a discrete prob
19、ability distribution It can be used to calculate the PMF for the number of occurrence of a particular event in a time or space interval (0, t) The occurrence of events are independent of each other Two or more events cannot occur simultaneouslyPMF of Poisson Distribution represents the mean occurren
20、ce rate of the event which is usually obtained from statistical data represents the number of occurrences of an event within a prescribed time (or space) interval (0, t)2 Common Probability Models2 Common Probability Models13132.8 Poisson Distribution Moments of Poisson DistributionThe Return Period
21、 of Poisson DistributionThe Annual Occurrence Probability of Poisson DistributionCDF of Poisson Distribution3 Random Vectors3 Random Vectors113.1 Definition of Random VectorsA random vector is defined as a vector (or set) of random variables3.2 The Joint CDF and PDF of Random VectorsThe Joint Cumula
22、tive Distribution FunctionThe Joint Probability Distribution FunctionFor continuous RVsFor discrete RVs3 Random Vectors3 Random Vectors223.3 Marginal Density Function of Random VectorsFor continuous random variables, a marginal density function (MDF) for each is defined as3.4 Cases of Joint CDF and
23、PDF of Two Continuous RVsThe Joint CDF of X and YThe Joint PDF of X and YThe MDF of X and Y3 Random Vectors3 Random Vectors333.5 Conditional Distribution Function of Random VectorsFor continuous random variables, the conditional distribution function for a random vector (X,Y) is defined as3.6 Statis
24、tical Independence of Random VectorsIf the random variables X and Y are statistical independent, then3 Random Vectors3 Random Vectors443.7 Correlation of Random Variables(1) Covariance of Two RVs(2) Coefficient of CorrelationThe formula of correlation coefficient(1)Let X and Y be two random variable
25、s with mean values and and standard deviations and . The covariance of X and Y is defined as (2) For two continuous variables X and Y3 Random Vectors3 Random Vectors55Properties of correlation coefficient(1)(2) The values of indicates the degree of linear dependence between the two random variables
26、X and YIf is close to 1, then X and Y are linearly related to each otherIf is close to 0, then X and Y are not linearly related to each otherDifference between Uncorrelated and Statistical IndependentX and Y are uncorrelatedX and Y are statistical independentStatistical independent is a much stronge
27、r statement than uncorrelatedstatistical independentuncorrelated3 Random Vectors3 Random Vectors66(3) Covariance Matrix of Random VectorsFor a random vector with n random variables, the covariance matrix is defined asThe matrix of correlation efficient is defined as3 Random Vectors3 Random Vectors77
28、Properties of and(1) Symmetric matrices(2) The diagonal terms(3) If all n random variables are uncorrelated, then3 Random Vectors3 Random Vectors88Statistical Estimate of the Correlation CoefficientAssume that there are n observations of variable X and n observations of variable Ysample meansample s
29、tandard deviationsample estimate of the correlation coefficient4 Functions of Random Variables4 Functions of Random Variables111. Linear Functions of Random Variableswhere, the are constants.Let Y be a linear function of random variables :Moments of Linear Functions of Random Variables4 Functions of
30、 Random Variables4 Functions of Random Variables22Variance of Linear Functions of Uncorrelated Random VariablesIf the n random variables are uncorrelated with each other, then for Properties of Linear Functions of Random Variables(1) The probability distributions of the random variables are not need
31、ed.(2) The linear function Y of uncorrelated normal random variables is a normal random variable with distribution parameters and .(3) The constant does not affect the variance, but it does affect the mean value.4 Functions of Random Variables4 Functions of Random Variables332. Product of Lognormal
32、Random VariablesLet Y be a function involving the products of several random variablesFor example,where K is a constant.Assume that these random variables are statistical independent , lognormal random variables. The above formula represents the sum of normally distributed random variables .The quan
33、tity is a normally distributed random variable is a lognormally distributed random variable4 Functions of Random Variables4 Functions of Random Variables55Moments of the lognormally distributed random variable Y4 Functions of Random Variables4 Functions of Random Variables663. Nonlinear Functions of
34、 Random VariablesLet Y be a general nonlinear function of the random variablesThe First Order Taylor Series Expansion of YMoments of Nonlinear Function Y. MathematicallyWhere, is called “design point” which is denoted by . for are uncorrelated .5 Central Limit Theorems5 Central Limit Theorems11Let t
35、he function Y be the sum of n statistically independent random variables whose probability distributions are arbitrary.The central limit theory states that as n approaches infinity, the sum of these independent random variables approaches a normal probability distribution if none of the random varia
36、bles tends to dominate the sum.AssumptionsTheoremIf we have a function defined as the sum of a large number of random variables, then we would expect the sum to be approximately as a normally distributed.ConclusionsThe sum of variables is often used to model the total load on a structure. Therefore,
37、 the total load can be approximated as a normal variable.1.Sum of Random Variables5 Central Limit Theorems5 Central Limit Theorems22Let Y be a product of n statistically independent random variables of the form:AssumptionsTransformation and TheoremIf we have a product of many independent random vari
38、ables, then the product approaches a lognormally distribution.ConclusionsThe product of variables is often used to model the resistance (or capacity) of a structure or structural component. Therefore, the resistance can be approximated as a lognormal variable.By using the central limit theorem, we c
39、an conclude that as n approaches infinity, approaches a normal probability distribution . If is normal, then Y must be lognormal.2. Product of Random VariablesHomework 1Plot the CDFs and PDFs of the following common random variables in the environment of MATLAB or EXCEL:(1)Uniform RV(2)Normal RV(3)Lognormal RV(4)Gamma RV(5)Extreme (6)Extreme (7)Extreme Note:(1)The parameters of these RVs can be assumed according to your willing.(2)These figures should be plotted by using numerical method.(3)These figures as well as their subroutines should be printed formally.Homework SetsHomework Sets
