Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 46 / 1

Write the partitioned mean as $({\mu }_1^T,{\mu }_2^T{)}^T$ and assume $V_{22}$ is nonsingular, as required by the displayed inverse. For the conditional multivariate normal distribution, form the residualA linear transformation of a multivariate normal distribution is again normal. Direct calculation givesThe jointly normal blocks $U$ and $X_2$ are therefore independent: their joint characteristic function has no cross term in its quadratic exponent and factors into the two marginal characteristic functions. Conditioning leaves the distribution of $U$ unchanged. Reconstructing $X_1$ proves the full conditional law, including the requested Schur complement covariance:This describes a regular conditional law; it does not require the equality event to have positive probability.

Put $D_1=X_1-Y_1$, $D_2=X_2-Y_2$ andThe difference vector is $D=BZ$. Its mean is zero because the two sons have the same mean vector. Its covariance matrix isFor example, $\mathrm{Cov}(D_1,D_2)=b-c-c+b=2(b-c)$. ThusThe positive definiteness assumption supplies $a-b>0$ and $a+b-2c>0$: these are eigenvalues of the original matrix on within-son contrast and between-son difference directions. In particular the resulting difference covariance is positive definite.

Taking the first marginal of the normal vector in part (i) givesEquivalently its variance is $a+a-2c$; the covariance between the two sons must be retained when taking their difference.

Applying the conditional multivariate normal distribution to the two differences gives, more generally,At $d=0$ the result isBoth this law and the marginal law have mean zero, but the conditional variance is smaller. Specifically, with $\rho =(b-c)/(a-c)$, the conditional variance is the marginal variance times $1-{\rho }^2$. The assumptions imply $0<\rho <1$, so the decrease is strict. Matching the breadth differences to zero gives information about the correlated length differences; it does not force the length differences themselves to be zero. The residual uncertainty remains positive. The original PDF has the between-son differences used here; the self-subtractions in the converted TeX are transcription errors.