Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.12 PROPERTIES OF JOINT DISTRIBUTIONS

More generally, we find (fora,bandcconstant)

V[aX+bY+c]=a^2 V[X]+b^2 V[Y]+2abCov[X, Y].

(30.136)

Note that ifXandYare in fact independent then Cov[X, Y] = 0 and we recover

the expression (30.68) in subsection 30.6.4.

We may use (30.136) to obtain an approximate expression forV[f(X, Y)]

for any arbitrary functionf, even when the random variablesXandY are

correlated. Approximatingf(X, Y) by the linear terms of its Taylor expansion

about the point (μX,μY), we have

f(X, Y)≈f(μX,μY)+

(

∂f

∂X

)

(X−μX)+

(

∂f

∂Y

)

(Y−μY),

(30.137)

where the partial derivatives are evaluated atX=μXandY=μY. Taking the

variance of both sides, and using (30.136), we find

V[f(X, Y)]≈

(

∂f

∂X

) 2

V[X]+

(

∂f

∂Y

) 2

V[Y]+2

(

∂f

∂X

)(

∂f

∂Y

)

Cov[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 iff(X, Y) is linear inXandY.

For several variablesXi,i=1, 2 ,...,n, we can define the symmetric (positive

definite)covariance matrixwhose elements are

Vij=Cov[Xi,Xj], (30.139)

and the symmetric (positive definite)correlation matrix

ρij=Corr[Xi,Xj].

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 to

V[f(X 1 ,X 2 ,...,Xn)]≈

∑

i

(

∂f

∂Xi

) 2

V[Xi]+

∑

i

∑

j=i

(

∂f

∂Xi

)(

∂f

∂Xj

)

Cov[Xi,Xj],

where the partial derivatives are evaluated atXi=μXi.