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# 数学代写|信息论代写Information Theory代考|ECET602 SMI, in General Is not Entropy. Entropy Is a Special Case of SMI

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## 数学代写|信息论代写Information Theory代考|SMI, in General Is not Entropy. Entropy Is a Special Case of SMI

Shannon defined a quantity $\mathrm{H}$ as a function of the entire distribution
$$H\left(p_1, \ldots, p_n\right)=-\sum p_i \log p_i$$
It is clear that for any given distribution $\left(p_1, \ldots, p_n\right)$ one can define the corresponding SMI. The definition of the function $H\left(p_1, \ldots, p_n\right)$ was also generalized to the case of a continuous random variable. In this case, the SMI is a functional defined for any distribution density $f(x)$ :

$$H[f(x)]=-\int f(x) \log f(x) d x$$
Clearly, this functional is not always a positive number and it is not always a finite quantity. Some of the mathematical problems in the definition of the SMI for the continuous case were discussed in Chap. 2 and Appendix C of Ben-Naim [1]. We shall sometimes use the notation $H\left(p_1, \ldots, p_n\right)$, sometimes we use the notation either $H(X)$ or SMI $(X)$, for the SMI of the random variable $X$.

## 数学代写|信息论代写Information Theory代考|The “Vennity” of Using Venn Diagrams in Representing Dependence Between Random Variables

Venn diagrams are useful for representing various operations between sets or events. In Fig. 1.2, we show a few examples of Venn diagram for the operations:
(a) union (or sum) of two events $A \cup B$
(b) intersection (or product) of two events $A \cap B$
(c) disjoint (or mutually exclusive) of two events $A \cap B=\phi$
(d) inclusion; $\mathrm{B}$ is contained in $\mathrm{A}, B \subset A$.
In using the Venn diagram, a region represents an event and the area of the region is proportional to the probability of that event. For instance; we throw a dart on a board, and we know that the dart hit the board. This means that the certainty event $\Omega$, which is the area of the entire board, is assigned the probability one. A region A represents the event: “the dart hit the region A.” The probability of this event is the area of the region A. The union of the two circles in Fig. 1.2a represents the probability that either the events A, or the event B have occurred (this is the set of all points belonging to $\mathrm{A}$ or $\mathrm{B}$, or both). The intersection of A and B in Fig. 1.2b represents the probability that both A and B have occurred (it contains all the points belonging to both A and B). Two disjoint events mean that there are no points belonging to both A and B, Fig. 1.2c. In this case, the probability of the event $A \cup B$ is simply the sum of the probabilities $P(A)$ and $P(B)$. In Fig. 1.2d the event B is contained in A, i.e. all points of B belong to A. The occurrence of B implies the occurrence of A. The occurrence of A does not necessarily imply the occurrence of B. In this case, the probability of B is smaller than the probability of A.

## 数学代写|信息论代写Information Theory代考|SMI, in General Is not Entropy. Entropy ls a Special Case of SMI

$$H\left(p_1, \ldots, p_n\right)=-\sum p_i \log p_i$$

$$H[f(x)]=-\int f(x) \log f(x) d x$$

## 数学代写|信息论代写Information Theory代考|The “Vennity” of Using Venn Diagrams in Representing Dependence Between Random Variables

(a) 两个事件的并集（或求和） $A \cup B$
(b) 两个事件的交集 (或积) $A \cap B$
(c) 两个事件不相交 (或相互排斥) $A \cap B=\phi$
（d）包容； $\mathrm{B}$ 包含在 $\mathrm{A}, \mathrm{B} \subset A$.

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