A.7 Functions of Random Variables in Visual Studio .NET

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Much work in probability theory involves manipulating functions of random variables. If some set of random variables has known distributions, it is desired to nd the distribution or the parameters of the distribution of a function of the random variables. 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]
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(A.40) (A.41) (A.42)
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E[a + bX] = a + bE[X]
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E[g1 (X) + g2 (X)] = E[g1 (X)] + E[g2 (X)]
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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. If
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When the variables are uncorrelated, the last equation simpli es to
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(A.46)
When variables are multiplied, the expected value of the result is E[XY ] = E[X]E[Y ] + Cov [X, Y ] (A.47)
FUNCTIONS OF RANDOM VARIABLES
However, only when the variables are uncorrelated is there a simple expression for the variance: 2 2 2 2 Var[XY ] = 2 Y + 2 X + X Y (A.48) X Y and
2 XY
(A.49)
Many applications of functions of random variables involve integrating the probability density functions. 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
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Reliability and Statistics in Geotechnical Engineering Gregory B. Baecher and John T. Christian 2003 John Wiley & Sons, Ltd ISBN: 0-471-49833-5