Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 47 / 2 / c

For common positive-definite covariance, subtracting the two Gaussian log densities cancels the quadratic term. Equal priors leave the linear discriminant analysis scoreHereThusAssign class two if the expression is negative; a tie has zero probability under either continuous Gaussian law.

To compute the error rather than just specify the boundary, put $D^2={\Delta }^T{\Sigma }^{-1}\Delta =27/2$. Within class one the score has mean $D^2/2=27/4$, and within class two mean $-27/4$. In both classes its variance is ${\Delta }^T{\Sigma }^{-1}\Sigma {\Sigma }^{-1}\Delta =D^2$. Hence both conditional error probabilities are $\Phi (-D/2)$, where $\Phi$ is the standard normal distribution function. Equal priors give the equal-covariance Gaussian classification errorFor the specified observation, the score is $\delta (2,1)=27/4>0$, so assign it to class one.