Let $S$ be a discrete random variable with the probability distribution:

$S=\{\begin{array}{ccc}1& w.p.& 0\mathrm{.}4\\ 2& w.p.& 0\mathrm{.}2\\ 3& w.p.& 0\mathrm{.}2\\ 4& w.p.& 0\mathrm{.}1\\ 5& w.p.& 0\mathrm{.}1\end{array}$

Consider a source code that maps Sin$\{1,2,3,4,5\}$ to codeword $f(S).$

$($a$)$ Calculate the entropy of the random variable $S.$

$($b$)$ Using the binary code tree that we learned in class $($Lecture $1),$ con$-$

struct a source code that minimizes the expected codeword length

$.$

$($c$)$ Compare the expected codeword length of your code with $H(S).$

Which one is smaller? Also explain why.