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--- tanszek:oktatas:techcomm:formulas_for_mathematical_exercises [2024/09/06 14:10] – [Combinatorics] kissa
+++ tanszek:oktatas:techcomm:formulas_for_mathematical_exercises [2024/10/15 18:47] (current) – [Information Theory] kissa
@@ Line 13: / Line 13: @@
 ==== Information Theory ====
-^ Notation  ^ Value  ^ Formula ^
+^ Notation     ^ Value                                                                                               ^ Formula                                                                                                                         ^
-| $$I(A)$$ | Information content or self-information of an event A. | $$I(A) = -\log_2 P(A)  \text{ [bits]}$$ |
+| $$I(A)$$     | Information content or self-information of an event A.                                              | $$I(A) = -\log_2 P(A) = \log_2 \frac{1}{P(A)}  \text{ [bits]}$$                                                                 |
-| $$H(X)$$ | Entropy, which measures the average amount of information (or uncertainty) in a random variable X. | $$H(X) = -\sum_{x \in X} P(x) \log_2 P(x)  \text{ [bits]}$$ |
+| $$H(X)$$     | Entropy, which measures the average amount of information (or uncertainty) in a random variable X.  | $$H(X) = -\sum_{x \in X} P(x) \log_2 P(x) = \sum_{x \in X} P(x) \log_2 \frac{1}{P(x)}  \text{ [bits]}$$                         |
-| $$H_{max}$$ | Maximum possible entropy (when all outcomes are equally likely). | $$H_{\text{max}} = \log_2 |\mathcal{X}|$$ $$|\mathcal{X}| \text{ is the number of possible outcomes in the set } \mathcal{X}$$ |
+| $$H_{max}$$  | Maximum possible entropy (when all outcomes are equally likely).                                    | $$H_{\text{max}} = \log_2 |\mathcal{X}|$$ $$|\mathcal{X}| \text{ is the number of possible outcomes in the set } \mathcal{X}$$  |
-| $$R(X)$$ | Redundancy, which measures the portion of duplicative information within a message. | $$R(X) = 1 - \frac{H(X)}{\log_2 |X|}$$  In terms of maximum entropy: $$R = \frac{H_{\text{max}} - H}{H_{\text{max}}}$$ |
+| $$R(X)$$     | Redundancy, which measures the portion of duplicative information within a message.                 | $$R = \frac{H_{\text{max}} - H}{H_{\text{max}}}$$          |
 ==== Combinatorics ====
@@ Line 23: / Line 23: @@
 | **Permutations** \\ number of all possible arrangements of $n$ elements | $$P_n = n!$$ | $$P_n^{k_1, k_2,...k_r} = \frac{n!}{k_1! \cdot k_2! \cdot ... \cdot k_r!}$$ |
 | **Variations** \\ the number of all possible arrangements of any $k$ elements from $n$ elements | $$V_n^k=\frac{n!}{(n-k)!}$$ | $$\overline{V}_n^k=n^k$$ |
-| **Combinations** \\ number of ways to choose $k$ items from $n$ items, regardless of order | $$C_n^k=\binom{n}{k}$$ | $$\overline{C}_n^k=\binom{n+k-1}{k}$$ |
+| **Combinations** \\ number of ways to choose $k$ items from $n$ items, regardless of order | $$C_n^k=\binom{n}{k}=\frac{n!}{k! \cdot (n - k)!}$$ | $$\overline{C}_n^k=\binom{n+k-1}{k}$$ |
 === What formula to use? ===
 | | |  **Repetition**   ||
-| | | //Not possible// | //Possible// |
+| | |  //Not possible//  |  //Possible//  |
-|| **Order** | //Matters// |  $$V_n^k$$ variation without repetition  |  $$\overline{V}_n^k$$ variation with repetition  |
+| **Order** | //Matters// |  $$V_n^k$$ (variation without repetition)  |  $$\overline{V}_n^k$$ (variation with repetition)  |
-|| ::: | //Doesn't matter// |  $$C_n^k$$ combination without repetition  |  $$\overline{C}_n^k$$ combination with repetition  |
+| ::: | //Doesn't matter// |  $$C_n^k$$ (combination without repetition)  |  $$\overline{C}_n^k$$ (combination with repetition)  |