PROBABILITY
and thus, using (30.135), we obtain
Cov[Xi,Xj]=E[(Xi−μi)(Xj−μj)] = (A−^1 )ij.HenceAis equal to the inverse of the covariance matrixVof theXi, see (30.139).
Thus, with the correct normalisation,f(x) is given by
f(x)=1
(2π)n/^2 (detV)^1 /^2exp[
−^12 (x−μ)TV−^1 (x−μ)]
.
(30.148)Evaluate the integralI=∫
∞exp[
−^12 (x−μ)TV−^1 (x−μ)]
dnx,whereVis a symmetric matrix, and hence verify the normalisation in (30.148).We begin by making the substitutiony=x−μto obtain
I=∫
∞exp(−^12 yTV−^1 y)dny.SinceVis a symmetric matrix, it may be diagonalised by an orthogonal transformation to
the new set of variablesy′=STy,whereSis the orthogonal matrix with the normalised
eigenvectors ofVas its columns (see section 8.16). In this new basis, the matrixVbecomes
V′=STVS=diag(λ 1 ,λ 2 ,...,λn),where theλiare the eigenvalues ofV. Also, sinceSis orthogonal, detS=±1, and so
dny=|detS|dny′=dny′.Thus we can writeIas
I=
∫∞
−∞∫∞
−∞···
∫∞
−∞exp(
−
∑ni=1y′i^2
2 λi)
dy′ 1 dy′ 2 ···dy′n=
∏ni=1∫∞
−∞exp(
−
yi′^2
2 λi)
dyi′=(2π)n/^2 (λ 1 λ 2 ···λn)^1 /^2 , (30.149)where we have used the standard integral
∫∞
−∞exp(−αy(^2) )dy=(π/α) 1 / (^2) (see subsection
6.4.2). From section 8.16, however, we note that the product of eigenvalues in (30.149) is
equal to detV. Thus we finally obtain
I=(2π)n/^2 (detV)^1 /^2 ,
and hence the normalisation in (30.148) ensures thatf(x) integrates to unity.
The above example illustrates some importants points concerning the multi-
variate Gaussian distribution. In particular, we note that theYi′areindependent
Gaussian variables with mean zero and varianceλi. Thus, given a general set of
nGaussian variablesxwith meansμand covariance matrixV, one can always
perform the above transformation to obtain a new set of variablesy′, which are
linear combinations of the old ones and are distributed as independent Gaussians
with zero mean and variancesλi.
This result is extremely useful in proving many of the properties of the mul-