The image you've provided contains a problem related to constructing Shannon-Fano and Huffman codes for a discrete source with given symbol probabilities. The task includes determining the maximum entropy $H_{max}$, the entropy of the source $H(X)$, the average number of bits per symbol, the compression coefficient $\mu$, and comparing the efficiency of the Shannon-Fano and Huffman codes.

Let's break down the steps needed to solve this:

1. Shannon-Fano Code Construction:

   • Use the probabilities provided in the table to generate a Shannon-Fano code.
   • The Shannon-Fano algorithm recursively divides the probabilities into two groups with equal or nearly equal sums, assigning binary codes accordingly.

2. Huffman Code Construction:

   • The Huffman coding algorithm builds an optimal binary tree, assigning shorter codes to more probable symbols.
   • We will use the same probabilities to construct a Huffman code.

3. Maximum Entropy $H_{max}$:

   • For a source with $N$ symbols, the maximum entropy occurs when all symbols are equally likely: $H_{max} = \log_2 N$

4. Source Entropy $H(X)$:

   • Entropy $H(X)$ is calculated using the formula: $H(X) = - \sum_{i=1}^{N} p_i \log_2 p_i$ where $p_i$ are the symbol probabilities from the table.

5. Average Number of Bits:

   • After generating the codes, the average number of bits per symbol $\overline{L}$ is given by: $\overline{L} = \sum_{i=1}^{N} p_i L_i$ where $L_i$ is the length of the code for the $i$-th symbol.

6. Compression Coefficient $\mu$:

   • The compression coefficient $\mu$ is the ratio of the source entropy to the average number of bits: $\mu = \frac{H(X)}{\overline{L}}$

7. Comparison of Efficiency:

   • Compare the efficiency of Shannon-Fano and Huffman codes by calculating and comparing their average code lengths and compression coefficients.

Would you like me to begin with any specific part of the problem or provide a step-by-step solution? Let me know if you need further clarification or a detailed explanation of any section.

Here are five related questions you might find interesting:

1. What is the key difference between Shannon-Fano and Huffman coding algorithms?
2. Why does Huffman coding generally produce more efficient codes than Shannon-Fano?
3. How does the source entropy relate to the efficiency of a code?
4. Can you think of a situation where the Shannon-Fano code could be more efficient than Huffman?
5. How do varying symbol probabilities affect the average number of bits per symbol?

Tip: The efficiency of a compression algorithm is directly related to how well the symbol probabilities are distributed—more uneven distributions tend to lead to higher compression.