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Questions and Answers
Match the following terms related to entropy with their definitions:
Match the following terms related to entropy with their definitions:
Entropy = Measure of uncertainty of a random variable H(X) = Entropy of a discrete random variable Distribution = How probabilities are assigned to random variable outcomes Logarithmic function = Mathematical function used to measure information content
Match the following entropy characteristics with their descriptions:
Match the following entropy characteristics with their descriptions:
H(X) ≥ 0 = Entropy is always non-negative Maximized entropy = Occurs when outcomes are uniformly distributed Log K = Maximum entropy for K values Zero entropy = Indicates no uncertainty in random variable outcomes
Match the following entropy components with their formulas:
Match the following entropy components with their formulas:
H(X) = $- rac{1}{n} imes ext{sum}(P(x) imes ext{log}(P(x)))$ P(x) = $ ext{Pr}ig{X=x\big ext{ for specific x}$ K values = Number of distinct outcomes in random variable p = Probability of a specific outcome in the random variable
Match the types of measurements of entropy with their units:
Match the types of measurements of entropy with their units:
Match the concepts of entropy with their effects or outcomes:
Match the concepts of entropy with their effects or outcomes:
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Study Notes
Entropy Function Overview
- Entropy measures the uncertainty of a random variable.
- For a discrete random variable X with alphabet x and probability mass function P(x) = Pr{X=x}, the probabilities describe the likelihood of outcomes.
Definition of Entropy
- The entropy H(X) is expressed mathematically:
H(X) = 1 * ∑ P(x) log_b(P(x)) for x ∈ X - Here, b denotes the logarithm base, which impacts the units of measurement.
Measurement Units
- Entropy reflects the average information contained in a random variable.
- Units of measurement include:
- Bits (base 2)
- Hartleys (base 10)
- Nats (base e)
Properties of Entropy
- Entropy is dependent solely on the distribution of X, not the specific values.
- The value of H(X) is always ≥ 0, indicating non-negative uncertainty.
Example of Entropy Calculation
- For a binary random variable X that takes values 0 with probability (1-p) and 1 with probability p, the entropy is given by: H(X) = -p log(p) - (1-p) log(1-p)
- Commonly represented as H(p, 1-p).
Maximum and Minimum Entropy
- Entropy reaches its maximum when p = 0.5 (equal uncertainty).
- H(X) is zero when p = 0 or p = 1, indicating no uncertainty about the outcome.
General Case with K Values
- When X can assume K values, the entropy is maximized when X is uniformly distributed across these values.
- In this uniform distribution case, entropy simplifies to:
H(X) = log(K)
Jensen's Inequality Application
- Using Jensen's inequality, it is established that:
H(X) ≤ log(Σ P(x)) for normalized probabilities. - This confirms that maximum entropy occurs with a uniform distribution, supporting H(X) ≤ log(K).
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