ASRAnetStructural Reliability & Risk Assessment4-8July2016Wuhan, ChinaLecture 1: Basic StatisticsProfessor Purnendu K. DasB.E., M.E., PhD, CEng, CMarEng, FRINA, FIStructE, FIMarEST
Structural Reliability & Risk Assessment 4 – 8 July 2016 Wuhan, China Lecture 1: Basic Statistics Professor Purnendu K. Das B.E., M.E., PhD, CEng, CMarEng, FRINA, FIStructE, FIMarEST 1
ASRAnet.DiscreteRandomVariableThe total number of occurrences is then given bym≥niN=7The mean value, uxand the standard deviation x of thediscrete variable xare calculated by1Anpx1VA[n;(,- uxOTheanalogybetweensecondmomentareaareproperty/masspropertycalculations and the determination of mean and standard deviationvalues (e.g.radius of gyration)maybenoted.2
• Discrete Random Variable The total number of occurrences is then given by The mean value, and the standard deviation 𝜎𝑥 of the discrete variable x are calculated by The analogy between second moment area are property/mass property calculations and the determination of mean and standard deviation values (e.g. radius of gyration) may be noted. m i i N n 1 x m i i x x i n N x m i i x i n N x 1 1 2 1 1 2
ASRAnetThe probability of occurrence of x, is given byMiDNTheprobability distribution p,as shown in Fig.1 has asimilarpatterntothatasfreguencydistributionshownin Fig 2. The total area under this curve is unity.mZP;3
The probability of occurrence of xi is given by The probability distribution pi as shown in Fig.1 has a similar pattern to that as frequency distribution shown in Fig 2. The total area under this curve is unity. N i n i p 1 1 m i i p 3
ASRANetmni=lLP:i=lPiX+X*2X*2XhFig 1.ProbabilityDistributionFig2.FrequencyDistribution
Fig 1. Probability Distribution Fig 2. Frequency Distribution 4
ASRAnet. ContinuousRandomVariableFor a random variable x which is continuous theproblem can be treated in a similar way by taking afinite range Ax as shown in Fig.3. The probabilitydistribution p(x),which is assumed tobe a continuouscurve,is givenbyn:p(x) = f(x)dx = limAxN△x →0andJtα f(x)dx = 15
• Continuous Random Variable For a random variable x which is continuous the problem can be treated in a similar way by taking a finite range as shown in Fig.3. The probability distribution p(x), which is assumed to be a continuous curve, is given by and −∝ +∝ 𝑓 𝑥 𝑑𝑥 = 1 0 ( ) ( ) lim x x N i n p x f x dx x 5
ASRANetpaf(n)dnf(n)f(n)dn =1-0Fig3.ProbabilityDistributionFig4.ProbabilityDensityFunction6
Fig 3. Probability Distribution Fig 4. Probability Density Function 6
ASRANetThis is the probability of occurrencex withindx. In this case f(x) is called the probability densityfunction Fig. 4The mean value μx and the standard deviation ox of acontinuous random variable x with density functionf(x) are calculated by;+8(1)μx=X=f(x)xdx8(2) = Jtα f(x) (x -μx)2dx
This is the probability of occurrence x within the band dx. In this case f(x) is called the probability density function Fig. 4 The mean value 𝜇𝑥 and the standard deviation 𝜎𝑥 of a continuous random variable x with density function f(x) are calculated by; = −∝ +∝ 𝑓 𝑥 (𝑥 − 𝜇𝑥) 2𝑑𝑥 x f x xdx x 7
ASRANetThe yariance js the second central moment of thearea of PDF with respect to its centre of gravity(again analogous to the radius of gyration). Thepositive square root of the variance is called thestandard deviation. The coefficient of variation, Vx is acharacteristicandenabiesadimensionlesscomparison to be made of a number of randomvariables of differentdimensions..It is definedas theratio of the standard deviationto the meanvalue of arandomvariablei.e.0xVx =μxTherearemanywellknowntheoreticallyestablishedprobabilitydensity functions which have a widevariety ofshapes.8
The variance is the second central moment of the area of PDF with respect to its centre of gravity (again analogous to the radius of gyration). The positive square root of the variance is called the standard deviation. The coefficient of variation, 𝑉𝑥 is a dimensionless characteristic and enables a comparison to be made of a number of random variables of different dimensions. It is defined as the ratio of the standard deviation to the mean value of a random variable i.e. 𝑉𝑥 = 𝜎𝑥 𝜇𝑥 There are many well known theoretically established probability density functions which have a wide variety of shapes. 8
ASRAnetAccording to the Central Limit Theorem thedistribution of the sum of n terms, each of which hasa finite variance, approaches a normal (Gaussian)distribution as n approaches infinity. The distributionof the sums converges very rapidly to a normal(Gaussian) distribution when the variances of theterms are of the same order of magnitude.When nexceeds about ten,thedifferencesbetween theresultant distribution and the normal distribution isnegligibleinmostcases.9
According to the Central Limit Theorem the distribution of the sum of n terms, each of which has a finite variance, approaches a normal (Gaussian) distribution as n approaches infinity. The distribution of the sums converges very rapidly to a normal (Gaussian) distribution when the variances of the terms are of the same order of magnitude. When n exceeds about ten, the differences between the resultant distribution and the normal distribution is negligible in most cases. 9
ASRAnetThe probability density function f(x) of a normaldistributiondefinedmathematicallybytwoisparameters μx and ox as (because symmetrical notethat the mean and the median are both the same):1f7r2元0xThe probability of occurrence for x < x, is given by thecumulative distribution function F(x) asX11pxdxf(x)dexp-21T2元0810
The probability density function f(x) of a normal distribution is defined mathematically by two parameters 𝜇𝑥 and 𝜎𝑥 as (because symmetrical note that the mean and the median are both the same): The probability of occurrence for x < x1 is given by the cumulative distribution function F(x) as 𝐹(𝑥1) = −∞ 𝑥1 𝑓 𝑥 𝑑𝑥 = 1 2𝜋𝜎𝑥 −∞ 𝑥1 𝑒𝑥𝑝 − 1 2 𝑥 − 𝜇𝑥 𝜎𝑥 2 𝑑𝑥 2 2 1 2 1 ( ) x x x exp x f x 10