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Виріши 12 номер завдання 3,4

To solve **task 12** for numbers 3 (productivity) and 4 (redundancy) using the provided data, we can follow these steps:

### 1. Given data for task 12:
From the table:
- Probabilities:  
  \( p(x_1) = 0.14, p(x_2) = 0.29, p(x_3) = 0.27, p(x_4) = 0.14, p(x_5) = 0.04, p(x_6) = 0.12 \)
  
- Symbol durations (in ms):  
  \( \tau_1 = 0.2, \tau_2 = 0.1, \tau_3 = 0.5, \tau_4 = 0.12, \tau_5 = 7.3, \tau_6 = 4.8 \)

### 2. Calculation of Productivity \( H \) and Redundancy \( R \):

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

Now, using the probabilities from task 12:

\[
H = -(0.14 \log_2 0.14 + 0.29 \log_2 0.29 + 0.27 \log_2 0.27 + 0.14 \log_2 0.14 + 0.04 \log_2 0.04 + 0.12 \log_2 0.12)
\]

Let's compute this.The entropy \( H \) for task 12 is approximately \( 2.375 \, \text{bit} \).

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

Now, let's calculate redundancy.It seems like I can't do more advanced data analysis right now. Please try again later.

Would you like me to help with any additional steps manually?

Calculate the productivity and redundancy for task 12 using probabilities 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, and p(x_6) = 0.12, along with symbol durations τ = [0.2, 0.1, 0.5, 0.12, 7.3, 4.8] ms. Learn how to apply the formulas for entropy, productivity, and redundancy based on information theory.

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