Podcast
Questions and Answers
What does the entropy of a random variable measure?
What does the entropy of a random variable measure?
Which formula correctly represents the entropy of a discrete random variable X?
Which formula correctly represents the entropy of a discrete random variable X?
When is the entropy maximized for a random variable X that can take on K values?
When is the entropy maximized for a random variable X that can take on K values?
What is the value of entropy H(X) when a random variable has a probability distribution of either P=0 or P=1?
What is the value of entropy H(X) when a random variable has a probability distribution of either P=0 or P=1?
Signup and view all the answers
What is the significance of the base 'b' in the entropy formula?
What is the significance of the base 'b' in the entropy formula?
Signup and view all the answers
Study Notes
Entropy Function
- Entropy measures the uncertainty of a random variable.
- For a discrete random variable ( X ) with alphabet ( x ) and probability mass function ( P(x) = \text{Pr}{X = x} ).
- Entropy ( H(X) ) is defined as: [ H(X) = -\sum_{x \in X} P(x) \log_b P(x) ]
- ( b ) denotes the base of the logarithm, influencing units of measurement (bits, Hartleys, or nats).
Characteristics of Entropy
- Represents the average information contained in a random variable ( X ).
- Depends solely on the distribution of ( X ), not the specific values ( X ) can take.
- Always greater than or equal to zero, ( H(X) \geq 0 ).
Example of Binary Entropy
- For random variable ( X ) taking values:
- ( 0 ) with probability ( 1-p )
- ( 1 ) with probability ( p )
- Entropy given by: [ H(X) = -p \log p - (1-p) \log (1-p) ]
- Denoted sometimes as ( H(p, 1-p) ).
- Maximum entropy occurs when ( p = 0.5 ), and entropy is zero when ( p = 0 ) or ( p = 1 ).
Uniform Distribution and Maximum Entropy
- For a random variable ( X ) taking ( K ) values, entropy is maximized when ( X ) is uniformly distributed among these values.
- In this case: [ H(X) = \log K ]
Calculating Entropy
- Scenarios where: [ H(X) = -\sum_{x \in X} P(x) \log \left(\frac{1}{P(x)}\right) ]
- Using Jensen's inequality for convex functions: [ \sum \lambda f(x) \leq f\left(\sum \lambda x\right) ]
- Resulting inequality shows: [ H(X) \leq \log K ]
- Indicates that maximum entropy ( H(X) ) occurs at ( \log K ) when uniformly distributed.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
This quiz covers the fundamentals of entropy in information theory, focusing on its definition and calculation involving discrete random variables. Participants will explore the concept of uncertainty and how it is quantified through the entropy function. Test your understanding of how probability mass functions relate to entropy.
