Math Problem Statement
Suppose S={s1, s2, s3, s4, s5, s6}, P(s1)=P(s2)= P(s3)=1/4, P(s4)=p(s5)=1/16, P(s6)=1/8. What is the entropy of S? (Use log2)
Solution
The entropy of a discrete random variable with probabilities is given by:
Using the given probabilities for each element in the set :
Now, we calculate the entropy:
Each logarithmic term can be simplified:
Now substitute these values back into the equation:
Simplify each term:
= - \left( -\frac{3}{2} + -\frac{1}{2} + -\frac{3}{8} \right)$$ Combining terms: $$H(S) = \frac{3}{2} + \frac{1}{2} + \frac{3}{8} = 2 + 0.375 = 2.375$$ Thus, the entropy of $$S$$ is: $$H(S) = 2.375 \, \text{bits}$$ Would you like further clarification on this or any specific step? Here are 5 related questions: 1. How does entropy change with more equally probable outcomes? 2. How is entropy used in information theory? 3. What happens to entropy if one outcome becomes much more probable than others? 4. Can entropy ever be negative? 5. How do you calculate entropy when probabilities are not powers of 2? **Tip:** Always double-check the probabilities sum to 1 to ensure correctness in entropy calculations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Information Theory
Entropy
Probability
Formulas
H(S) = - ∑ P(si) * log2(P(si))
Theorems
Shannon Entropy
Suitable Grade Level
Undergraduate Level / Advanced High School
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