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1、Elements of Probability Theory,Dr.Dagang Lu,ProfessorSchool of Civil EngineeringHarbin Institute of Technology,1 Random Variables 1,1.1 Definition of Random Variables,Let be a random experiment,is its sample space,for,there exists a real single-value function.,If for,is a random event on event field
2、,then we can call a random variable.,or,1.2 Description of Random Variables,1.Probability Mass Function(PMF)for discrete random variable,There are three kinds of functions for description of random variables as follows:,2.Cumulative Distribution Function(CDF)for discrete random variablefor continuou
3、s random variable,3.Probability Density Function(PDF)for continuous random variable,1 Random Variables 2,1 Random Variables 3,1.Probability Mass Function(PMF),The probability mass function(PMF)is defined for discrete random variables as follows:represents probability that a discrete random variable
4、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 functions(continuous and discrete)corresponding to values less than or equal to.Mathematically,The cumulative distribution function(CDF)i
5、s defined for both discrete and continuous random variables as follows:,1 Random Variables 4,3.Probability Density Function(PDF),For continuous random variables,the probability density function(PDF)is defined as the first derivative of the cumulative function.Mathematically,Properties of CDF,PDF and
6、 PMF,The CDF is a positive,nondecreasing function whose value is between 0 and 1:,If,then,For continuous random variable,1 Random Variables 5,1.3 Moments of Random Variables,1.Mean or Expected Value(First Moment),The mean value of a random variable is denoted by,For a continuous random variable,the
7、mean value is defined as,For a discrete random variable,the mean value is defined as,The expected value of is commonly denoted by and is equal to the mean value of,1 Random Variables 6,For a discrete random variable,the nth moment is defined as,For a continuous random variable,the nth moment is defi
8、ned as,The expected value of is called the nth moment of,The mathematical expectation of an arbitrary function of the random variable is defined as,1 Random Variables 7,The standard deviation of is defined as the positive square root of the variance:,An important formula,The variance of a random var
9、iable is a measure of the degree of randomness about the mean value:,2.Variance and Standard Deviation(Second Moment),1 Random Variables 8,If 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 d
10、ivided by the mean:,3.Moments of Sample,the true mean can be approximated by the sample mean,the true standard deviation can be approximated by the sample standard deviation,1 Random Variables 9,1.4 Standard Form of Random Variables,Let be a random variable.The standard form of,denoted by,is defined
11、 as,the mean value of is calculated as follows:,the deviation of is calculated as follows:,2 Common Probability Models1,2.1 Uniform Random Variables,2 Common Probability Models2,2.2 Normal Random Variables,For a standard normal RV U,For a general normal RV X,The PDF of U is denoted by,The CDF of U i
12、s denoted by,2 Common Probability Models3,Standard Normal Random Variables,Relationship between general normal RV X and standard normal RV U,2 Common Probability Models4,Properties of distribution function of a normal RV,The PDF is symmetrical about the mean value,The sum of and is equal to 1,The in
13、verse CDF of a normal RV,2 Common Probability Models5,2.3 Lognormal Random Variables,CDF&PDF lognormal RVs,Definition of Lognormal RVs,2 Common Probability Models6,Moments of Lognormal RVs,If,then,Properties of Lognormal RVs,2 Common Probability Models7,2.4 Gamma Distribution,PDF of Gamma RVs,Gamma
14、Function,for,Moments of Gamma RVs,are distribution parameters,2 Common Probability Models8,2.5 Extreme Type(Gumbel Distribution),CDF&PDF of Extreme RVs,for,Moments of Extreme RVs,are distribution parameters,2 Common Probability Models9,2.6 Extreme Type,CDF&PDF of Extreme RVs,for,Moments of Extreme R
15、Vs,are distribution parameters,2 Common Probability Models10,2.7 Extreme Type(Weibull Distribution),CDF of the Largest Values,for,Moments of the Largest Values,are distribution parameters,2 Common Probability Models11,2.7 Extreme Type(Weibull Distribution),CDF of the Smallest Values,for,Moments of t
16、he Smallest Values,are distribution parameters,2 Common Probability Models12,2.8 Poisson Distribution,Properties of Poisson Distribution,Assumptions of Poisson Distribution,It is a discrete probability distribution It can be used to calculate the PMF for the number of occurrence of a particular even
17、t in a time or space interval(0,t),The occurrence of events are independent of each other Two or more events cannot occur simultaneously,PMF of Poisson Distribution,represents the mean occurrence rate of the event which is usually obtained from statistical data,represents the number of occurrences o
18、f an event within a prescribed time(or space)interval(0,t),2 Common Probability Models13,2.8 Poisson Distribution,Moments of Poisson Distribution,The Return Period of Poisson Distribution,The Annual Occurrence Probability of Poisson Distribution,CDF of Poisson Distribution,3 Random Vectors1,3.1 Defi
19、nition of Random Vectors,A random vector is defined as a vector(or set)of random variables,3.2 The Joint CDF and PDF of Random Vectors,The Joint Cumulative Distribution Function,The Joint Probability Distribution Function,For continuous RVs,For discrete RVs,3 Random Vectors2,3.3 Marginal Density Fun
20、ction of Random Vectors,For continuous random variables,a marginal density function(MDF)for each is defined as,3.4 Cases of Joint CDF and PDF of Two Continuous RVs,The Joint CDF of X and Y,The Joint PDF of X and Y,The MDF of X and Y,3 Random Vectors3,3.5 Conditional Distribution Function of Random V
21、ectors,For continuous random variables,the conditional distribution function for a random vector(X,Y)is defined as,3.6 Statistical Independence of Random Vectors,If the random variables X and Y are statistical independent,then,3 Random Vectors4,3.7 Correlation of Random Variables,(1)Covariance of Tw
22、o RVs,(2)Coefficient of Correlation,The formula of correlation coefficient,(1),(2)For two continuous variables X and Y,3 Random Vectors5,Properties of correlation coefficient,(1),(2)The values of indicates the degree of linear dependence between the two random variables X and Y,If is close to 1,then
23、 X and Y are linearly related to each other,If is close to 0,then X and Y are not linearly related to each other,Difference between Uncorrelated and Statistical Independent,X and Y are uncorrelated,X and Y are statistical independent,Statistical independent is a much stronger statement than uncorrel
24、ated,3 Random Vectors6,(3)Covariance Matrix of Random Vectors,For a random vector with n random variables,the covariance matrix is defined as,The matrix of correlation efficient is defined as,3 Random Vectors7,Properties of and,(1)Symmetric matrices,(2)The diagonal terms,(3)If all n random variables
25、 are uncorrelated,then,3 Random Vectors8,Statistical Estimate of the Correlation Coefficient,Assume that there are n observations of variable X and n observations of variable Y,sample mean,sample standard deviation,sample estimate of the correlation coefficient,4 Functions of Random Variables1,1.Lin
26、ear Functions of Random Variables,where,the are constants.,Let Y be a linear function of random variables:,Moments of Linear Functions of Random Variables,4 Functions of Random Variables2,Variance of Linear Functions of Uncorrelated Random Variables,If the n random variables are uncorrelated with ea
27、ch other,then,for,Properties of Linear Functions of Random Variables,(1)The probability distributions of the random variables are not needed.,(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
28、 the variance,but it does affect the mean value.,4 Functions of Random Variables3,2.Product of Lognormal Random Variables,Let Y be a function involving the products of several random variables,Assume that these random variables are statistical independent,lognormal random variables.,The above formul
29、a represents the sum of normally distributed random variables.,The quantity is a normally distributed random variable,is a lognormally distributed random variable,4 Functions of Random Variables5,Moments of the lognormally distributed random variable Y,4 Functions of Random Variables6,3.Nonlinear Fu
30、nctions of Random Variables,Let Y be a general nonlinear function of the random variables,The First Order Taylor Series Expansion of Y,Moments of Nonlinear Function Y,.Mathematically,Where,is called“design point”which is denoted by.,for are uncorrelated.,5 Central Limit Theorems1,Let the function Y
31、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 variables tends to
32、dominate the sum.,Assumptions,Theorem,If 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.,Conclusions,The sum of variables is often used to model the total load on a structure.Therefore,the total l
33、oad can be approximated as a normal variable.,1.Sum of Random Variables,5 Central Limit Theorems2,Let Y be a product of n statistically independent random variables of the form:,Assumptions,Transformation and Theorem,If we have a product of many independent random variables,then the product approach
34、es a lognormally distribution.,Conclusions,The 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 can conclude that as n approaches i
35、nfinity,approaches a normal probability distribution.If is normal,then Y must be lognormal.,2.Product of Random Variables,Homework 1,Plot the CDFs and PDFs of the following common random variables in the environment of MATLAB or EXCEL:,Uniform RVNormal RVLognormal RVGamma RVExtreme Extreme Extreme,Note:,The parameters of these RVs can be assumed according to your willing.These figures should be plotted by using numerical method.These figures as well as their subroutines should be printed formally.,Homework Sets,
