$2\mathrm{.}12$ Consider a two$-$class, two$-$dimensional classification task, where the feature

vectors in each of the classes ${}_1,{}_2$ are distributed according to

$({}_1|)({}_2|)=\frac{1}{\sqrt[\phantom{2}]{2{}_2^2}}$exp$(-\frac{1}{2{}_2^2}(x-{}_2{)}^T(x-{}_2))$

with

${}_1=[1,1{]}^T,{}_2=[1\mathrm{.}5,1\mathrm{.}5{]}^T,{}_1^2={}_2^2=0\mathrm{.}2$

Assume that $P({}_1)=P({}_2)$ and design a Bayesian classifier

$($a$)$ that minimizes the error probability

$($b$)$ that minimizes the average risk with loss matrix

$=\left[\begin{array}{cc}0& 1\\ 0\mathrm{.}5& 0\end{array}\right]$

Using a pseudorandom number generator, produce $100$ feature vectors from

each class, according to the preceding pdfs$.$ Use the classifiers designed to

classify the generated vectors. What is the percentage error for each case?

Repeat the experiments for ${}_2=[3\mathrm{.}0,3\mathrm{.}0{]}^T.$