$1.$ Evolutionary Computation $(20$ points rescaled to $5)$

Given the function $\backslash ($ f$($x$)=$x$^\{2\}-2$ x$+1\backslash ),$ we will use an evolutionary algorithm to find an approximation for the maximum value of the function for integers in the domain $[0,31].$

The solutions are represented using binary encoding.

a$.$ In Table $1,$ the initial population is provided. Compute the integer values, the function values, and fitness probabilities. Further, compute the total and average values. $(5$ points$)$

Table $1$: Genetic Algorithm $-$ Initial Population

b$.$ Draw the cumulative distribution function $($CDF$)$ in Figure $1$ for roulette wheel selection $($a sketch suffices$).$ For the CDF$,$ maintain the order of the chromosomes in Table $1.$ Given a random number $0\mathrm{.}32,$ which chromosome $($ID$)$ would be selected for roulette wheel selection? Indicate using lines. $(5$ points$)$