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Some simple results having wide application are as follows (X and Y are random variables; a and b are constants, g(X) is a function of X): E[aX] = aE[X] Var[X] = a 2 Var[X] Var[a + bX] = b Var[X]UPCA Encoder In VS .NETUsing Barcode creator for Visual Studio .NET Control to generate, create UPC A image in VS .NET applications.(A.40) (A.41) (A.42)UPCA Scanner In .NETUsing Barcode scanner for .NET Control to read, scan read, scan image in .NET framework applications.E[a + bX] = a + bE[X]Encoding Barcode In .NETUsing Barcode creator for Visual Studio .NET Control to generate, create bar code image in .NET framework applications.E[g1 (X) + g2 (X)] = E[g1 (X)] + E[g2 (X)]Bar Code Decoder In VS .NETUsing Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications.The variance of the sum of two functions is not so simple, as the variance is not a linear function. The linear combination of several random variables yields the following simple results. 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The most common examples are calculations of the moments of a joint probability distribution. From calculus we know that, if there is a set of variables xi that can be expressed in terms of another set yi and we want to integrate some function g(xi ), it may be more convenient to express the integral in terms of the yi variables. However, it is then necessary to transform the dxi into the dyi , and this requires using the Jacobian determinant. The equation for two variables in each set is g(xi , x2 ) dx1 dx2 = The Jacobian determinant is x1 (x1 , x2 ) y1 = x2 (y1 , y2 ) y1 x1 y2 x2 y2 g[x1 (y1 , y2 ), x2 (y1 , y2 )] (x1 , x2 ) dy1 dy2 (y1 , y2 ) (A.50)(A.51)For example, suppose that X1 and X2 are independent and uniformly distributed between 0 and 1. De ne Y1 = Y2 = It follows that 1 X1 = exp (Y1 + Y2 )2 2 X2 = 1 Y2 arctan 2 Y1 (A.53) 2 ln X1 cos 2 X2 2 ln X1 sin 2 X2 (A.52)If we evaluate the partial derivatives of the X s and substitute them into Equation (A.51), we nd 1 (X1 , X2 ) 1 2 2 = e Y1 /2 e Y2 /2 (A.54) (Y1 , Y2 ) 2 2 Thus, 1 2 dX1 dX2 = e Y1 /2 2 1 2 e Y2 /2 dY1 dY 2 2 (A.55)APPENDIX A: A PRIMER ON PROBABILITY THEORY which demonstrates that Equations (A.52) transform the uniform variables into independent standard Normal variables N(0.1). This result is useful in generating random Normal variables for Monte Carlo simulation. Extension of the Jacobian determinant for more than two variables is straightforward.References AASHTO (1994). LRFD Bridge Design Speci cations, SI Units. Washington, DC, American Association of State Highway and Transportation Of cials. AASHTO (1997). 1997 Interims to LRFD Highway Bridge Design Speci cations, SI Units, First Edition (1997 LRFD Interims). Washington, DC, American Association of State Highway and Transportation Of cials. AASHTO (1998). Standard Speci cations for Highway Bridges. Washington, DC, American Association of State Highway and Transportation Of cials. Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. New York, Dover Publications. ACI (2002). Building Code Requirements for Structural Concrete and Commentary (318-02 Building Code). Farmington Hills, MI, American Concrete Institute. Adler, R. J. (1981). The Geometry of Random Fields. Chichester, New York, John Wiley & Sons. Agterberg, F. P. (1974). Geomathematics. Mathematical Background and Geo-Science Applications. Amsterdam, New York, Elsevier. AISC (1994). Load and Resistance Factor Design. Manual of Steel Construction. Chicago, American Institute of Steel Construction. Aitchison, J. (1970). Choice against Chance: An Introduction to Statistical Decision Theory. Reading, MA, Addison-Wesley. Aitchison, J. and Brown, J. A. C. (1969). The Lognormal Distribution, with Special Reference to Its Uses in Economics. Cambridge, Cambridge University Press. Aitchison, J. and Dunsmore, I. R. (1975). Statistical Prediction Analysis. Cambridge, Cambridge University Press. Alaszewski, A., Harrison, L. and Manthorpe, J. (1998). Risk, Health, and Welfare: Policies, Strategies, and Practice. Philadelphia, PA, Open University Press. Allais, M. (1957). Methods of appraising economic prospects of mining exploration over large territories. Management Science 3: 285 347. Allen, D. E. (1994). The history and future of limit states design. Journal of Thermal Insulation and Building Envelopes 18: 3 20. Alloy, L. B. and Tabachnik, N. (1984). Assessment of covariation by humans and animals: the joint in uence of prior expectations and current situational information. Psychological Review 91: 112 149. Alpert, M. and Raiffa, H. (1982). A progress report on the training of probability assessors. Judgment Under Uncertainty, Heuristics and Biases. Kahneman, D., Slovic, P. and Tversky, A., eds., Cambridge, Cambridge University Press: 294 306. ANCOLD (1994). Guidelines on Risk Assessment 1994. Australia-New Zealand Committee on Large Dams. Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis. New York, John Wiley & Sons.Reliability and Statistics in Geotechnical Engineering Gregory B. Baecher and John T. Christian 2003 John Wiley & Sons, Ltd ISBN: 0-471-49833-5