$2.($a$)$ A memoryless information source $\backslash ($ M $\backslash )$ generates one of two messages every microsecond. The probabilities of these messages, $\backslash ($ m$\_\{1\}\backslash )$ and $\backslash ($ m$\_\{2\}\backslash ),$ are $0\mathrm{.}70$ and $0\mathrm{.}3,$ respectively.

$($i$)$ What is the source entropy of this source? Compute its information rate.

$($ii$)$ Find the optimum binary code for M$.$ Determine its code efficiency.

$($iii$)$ Suppose that the information source $\backslash ($ M $\backslash )$ is extended to second order. Find the optimum binary code and its code efficiency.

$(14$ Marks$)$