Mathematical Methods for Physics and Engineering - Matematica.NET

30.12 PROPERTIES OF JOINT DISTRIBUTIONSMore generally, we find (for a, b and c constant)V [aX + bY + c] =a 2 V [X]+b 2 V [Y ]+2ab Cov[X,Y ].(30.136)Note that if X and Y are in fact independent then Cov[X,Y ] = 0 and we recoverthe expression (30.68) in subsection 30.6.4.We may use (30.136) to obtain an approximate expression for V [ f(X,Y )]for any arbitrary function f, even when the random variables X and Y arecorrelated. Approximating f(X,Y ) by the linear terms of its Taylor expansionabout the point (µ X ,µ Y ), we have( )( )∂f∂ff(X,Y ) ≈ f(µ X ,µ Y )+ (X − µ X )+ (Y − µ Y ),∂X∂Y(30.137)where the partial derivatives are evaluated at X = µ X and Y = µ Y . Taking thevariance of both sides, and using (30.136), we find( ) 2 ( ) 2 ( )( )∂f∂f∂f ∂fV [ f(X,Y )] ≈ V [X]+ V [Y ]+2Cov[X,Y ].∂X∂Y∂X ∂Y(30.138)Clearly, if Cov[X,Y ] = 0, we recover the result (30.69) derived in subsection 30.6.4.We note that (30.138) is exact if f(X,Y ) is linear in X and Y .For several variables X i , i =1, 2,...,n, we can define the symmetric (positivedefinite) covariance matrix whose elements areand the symmetric (positive definite) correlation matrixV ij =Cov[X i ,X j ], (30.139)ρ ij =Corr[X i ,X j ].The diagonal elements of the covariance matrix are the variances of the variables,whilst those of the correlation matrix are unity. For several variables, (30.138)generalises toV [f(X 1 ,X 2 ,...,X n )] ≈ ∑ i( ∂f∂X i) 2V [X i ]+ ∑ i∑j≠i( ∂f∂X i)( ∂f∂X j)Cov[X i ,X j ],where the partial derivatives are evaluated at X i = µ Xi .1203