What are the most important formulas of the mathematical theory of communication?
The mathematical theory of communication, also known as information theory, was founded by Claude Shannon in his groundbreaking 1948 paper, "A Mathematical Theory of Communication." Here are some of the most important concepts and formulas:
1. **Entropy (H(X))**: The measure of the uncertainty of a random variable. For a discrete random variable X, the entropy is calculated as:
`H(X) = - ∑ P(x) log₂P(x)`
where the summation is over all possible outcomes x, and P(x) is the probability of outcome x. The base of the logarithm is 2 if the measure of information is in bits.
2. **Joint Entropy (H(X, Y))**: The measure of the combined uncertainty of two random variables X and Y:
`H(X, Y) = - ∑ ∑ P(x,y) log₂P(x,y)`
where the double summation is over all possible outcomes of X and Y, and P(x, y) is the joint probability of X and Y.
3. **Conditional Entropy (H(X|Y))**: The measure of the uncertainty of X given that Y is known:
`H(X|Y) = H(X, Y) - H(Y)`
4. **Mutual Information (I(X; Y))**: The measure of the amount of information that knowing the outcome of X gives about Y and vice versa:
`I(X; Y) = H(X) - H(X|Y) = H(Y) - H(Y|X)`
It can also be interpreted as the reduction in the uncertainty of one random variable due to the knowledge of another.
5. **Channel Capacity (C)**: The maximum rate at which information can be transmitted over a channel with a certain noise level:
`C = B log₂(1 + SNR)`
where B is the bandwidth of the channel, and SNR is the Signal-to-Noise Ratio. This formula is known as the Shannon Capacity Formula.
Remember, these formulas are part of Shannon's information theory, which is a foundational concept for much of modern telecommunications, computer science, and data science.
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