A discrete memoryless source has an alphabet A $=\{$a$,$ b$,$ c$,$ d$,$ e$,$ f$\}$ with

symbol probabilities P$($X$)=\{0\mathrm{.}1,0\mathrm{.}1,0\mathrm{.}1,0\mathrm{.}4,0\mathrm{.}2,0\mathrm{.}1\}.$

i$)$ Construct a binary Huffman code with minimum variance for this

source. $(2$ points$)$

ii$)$ Calculate the entropy of the source. $(1$ point$)$

iii$)$ Calculate the average codeword length of the Huffman code.

$(1$ point$)$

iv$)$ Calculate the coding efficiency of the Huffman code. $(1$ point$)$

v$)$ What is the variance of the codeword length for this constructed

Huffman code? $(3$ points$)$