https://file.aiquickdraw.com/mathbot/file/17262150956152v1k73ah.jpg

In the provided image, for variant 12, the table gives the probabilities \( p(x_1), p(x_2), p(x_3), p(x_4), p(x_5), p(x_6) \) and the symbol durations \( \tau_1, \tau_2, \tau_3, \tau_4, \tau_5, \tau_6 \). For this specific variant:

- \( p(x_1) = 0.14 \)
- \( p(x_2) = 0.29 \)
- \( p(x_3) = 0.27 \)
- \( p(x_4) = 0.14 \)
- \( p(x_5) = 0.04 \)
- \( p(x_6) = 0.12 \)

- \( \tau_1 = 0.2 \) ms
- \( \tau_2 = 0.1 \) ms
- \( \tau_3 = 0.5 \) ms
- \( \tau_4 = 1.5 \) ms
- \( \tau_5 = 7.3 \) ms
- \( \tau_6 = 4.8 \) ms

To solve the problem of calculating entropy \( H \), average duration \( \tau \), productivity \( H/\tau \), and redundancy \( R \), we will proceed step by step.

### 1. Entropy \( H \)
The entropy formula is:
\[
H = -\sum_{i=1}^{6} p(x_i) \log_2(p(x_i))
\]

### 2. Average Duration \( \tau \)
The average symbol duration is calculated as:
\[
\tau = \sum_{i=1}^{6} p(x_i) \tau_i
\]

### 3. Productivity
Productivity is:
\[
\text{Productivity} = \frac{H}{\tau}
\]

### 4. Redundancy
Redundancy is:
\[
R = 1 - \frac{H}{\log_2 M}
\]
where \( M = 6 \) (the size of the alphabet).

I will now calculate these values.

Calculate the entropy, average symbol duration, productivity, and redundancy for a non-Markov discrete information source with alphabet X of power M=6 using variant 12. This problem focuses on probabilities and symbol durations provided in the table.

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