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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.

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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.

Побудувати код Шеннона-Фано та код Гаффмана для дискретного джерела, символи якого мають задані у таблиці ймовірності. Визначити максимальну ентропію Hmax, ентропію джерела H(X), середнє число біт на n сер, коефіцієнт стиснення μ. Порівняти ефективність кодів Шеннона-Фано та Гаффмана.

Learn how to construct Shannon-Fano and Huffman codes for a discrete source with given probabilities. This problem involves calculating the maximum entropy (Hmax), source entropy (H(X)), average number of bits per symbol, and compression coefficient (μ). Compare the efficiency of Shannon-Fano and Huffman coding.

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