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