Math Problem Statement

Виріши 12 номер завдання 3,4

Виріши 12 номер завдання 3,4

Solution

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(x1)=0.14,p(x2)=0.29,p(x3)=0.27,p(x4)=0.14,p(x5)=0.04,p(x6)=0.12p(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):
    τ1=0.2,τ2=0.1,τ3=0.5,τ4=0.12,τ5=7.3,τ6=4.8\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 HH and Redundancy RR:

3. Productivity HH:

The formula for entropy HH is: H=i=16p(xi)log2p(xi)H = -\sum_{i=1}^{6} p(x_i) \log_2 p(x_i)

Now, using the probabilities from task 12:

H=(0.14log20.14+0.29log20.29+0.27log20.27+0.14log20.14+0.04log20.04+0.12log20.12)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 HH for task 12 is approximately 2.375bit2.375 \, \text{bit}.

4. Redundancy RR:

The formula for redundancy RR is: R=1Hlog2MR = 1 - \frac{H}{\log_2 M} where M=6M = 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?

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Math Problem Analysis

Mathematical Concepts

Information Theory
Entropy
Redundancy
Productivity

Formulas

Entropy formula: H = -Σ(p(x_i) * log_2(p(x_i)))
Productivity formula: H/τ
Redundancy formula: R = 1 - H/log_2(M)

Theorems

Shannon's Entropy Theorem

Suitable Grade Level

University