= Solution
Use a <multivariate normal distribution> within each species, a common positive-definite <covariance matrix>, equal <prior odds> and equal misclassification costs. Assume the training observations are independent and representative of the populations. Substituting the <sample means> and pooled <sample covariance matrix> gives a fitted <linear discriminant analysis> rule, rather than a rule with known population parameters.
The determinant of the pooled covariance is $8$, so
$$
S^{-1}=\frac18\begin{pmatrix}4&2\\2&3\end{pmatrix},\qquad
L=S^{-1}(\bar x_f-\bar x_a)=\begin{pmatrix}1/2\\-5/4\end{pmatrix}.
$$
The midpoint of the <sample means> is $(4,6)^T$, whose scalar product with $L$ is $-11/2$. Hence the fitted log <likelihood ratio> is
$$
\boxed{q(x)=\frac12x_1-\frac54x_2+\frac{11}{2}=\frac{2x_1-5x_2+22}{4}.}
$$
\b[Assign fattus when $q(x)>0$ and apathus when $q(x)<0$; either assignment is optimal on $q(x)=0$ under equal priors.] The separating line is $x_2=0.4x_1+4.4$, with fattus below it. As a sign check, the scores at the two <sample means> are $19/4$ and $-19/4$, respectively. This uses the corrected <Gaussian Bayes classifier> from (a).
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