A random variable $A$ takes values in the alphabet $\mathcal{A}=\{a, b, c, d, e\}$ with probabilities $0.4,0.2,0.2,0.1$ and $0.1$. Calculate the entropy of $A$.

Define what it means for a code for a general finite alphabet to be optimal. Find such a code for the distribution above and show that there are optimal codes for this distribution with differing lengths of codeword.

[You may use any results from the course without proof. Note that $\log _{2} 5 \simeq 2.32$.]