A source has an alphabet $\{$a$1,$ a$1,$ a$3,$ a$4\}$ with corresponding probabilities $\{0.1,$

$0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4\}.$

$1.$ Find the entropy of the source.

$2.$ What is the minimum required average code word length to represent this source

for error$-$free reconstruction?

$3.$ Design a Huffman code for the source and compare the average length of the

Huffman code with the entropy of the source.

$4.$ Design a Huffman code for the second extension of the source $($take two letters

at a time$).$ What is the average code word length? What is the average number

of required binary letters per each source output letter?

$5.$ Which is a more efficient coding scheme: the Huffman coding of the original

source or the Huffman coding of the second extension of the source?