$(30$ points$)$ We want to build a pattern classifier with continuous attribute using Bayes' Theorem. The object to be classified has one feature, $x$ in the range The conditional probability density functions for each class are listed below:

$P(x|C_1)=\{\begin{array}{ccc}\frac{1}{8}& if& 1x<mn\; id="EditorElement\_1861432">9</mn>\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \\ P(x|C_2)=\{\begin{array}{ccc}\frac{1}{9}(x-2)& if& 2x<mn\; id="EditorElement\_1861529">5</mn>\\ \frac{1}{9}(8-x)& if& 5x<mn\; id="EditorElement\_1861574">8</mn>\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}\end{array}$

$($a$)$ Assuming equal priors, $P(C_1)=P(C_2)=0\mathrm{.}5,$ classify an object with the attribute value $x=7.$

$($b$)$ Assuming unequal priors, $P(C_1)=0\mathrm{.}6,P(C_2)=0\mathrm{.}4,$ classify an object with the attribute value $x=4.$