### 经济代写|博弈论代写Game Theory代考|ECON3050

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Numbers and algebra

The set $\mathbb{R}$ of real numbers has an algebraic structure under the usual addition and multiplication rules for real numbers. $\mathbb{R}$ contains the set of natural numbers
$$\mathbb{N}={1,2, \ldots, n, \ldots} .$$
The computational rules of $\mathbb{R}$ may also be applied to $\mathbb{N}$ because sums and products of two natural numbers yield natural numbers. ${ }^{7}$ Similar algebraic rules can be defined on other sets. We give two examples below.

REMARK 1.6. There is the philosophical issue whether “0” is a natural number, which corresponds to the question whether an entity can be a “set” when it is empty, i.e., contains no element. ${ }^{8}$ For clarification, we therefore employ the notation
$$\mathbb{N}_{0}=\mathbb{N} \cup{0}$$
for the set of natural numbers including 0 .
Complex numbers. There is no real number $r \in \mathbb{R}$ with the property $r^{2}=-1$. To remedy this deficiency, one may introduce a new “number” i and do computations with it like it were a real number with the property
$$\mathrm{i}^{2}=-1 .$$

In doing so, one arrives at more general numbers of the form $z=$ $a+\mathrm{i} b$, with $a$ and $b$ being real numbers. The set
$$\mathbb{C}={a+\mathrm{i} b \mid a, b \in \mathbb{R}}$$
is the set of complex numbers. The special number
$$\mathrm{i}=0+\mathrm{i} \cdot 1$$
is the so-called imaginary unit. Using the algebraic rules of $\mathbb{R}$ and always keeping $\mathrm{i}^{2}=-1$ in mind, one can perform the usual computations with additions, subtractions, multiplications and divisions in $\mathbb{C}$

## 经济代写|博弈论代写Game Theory代考|Probabilities, information and entropy

Consider $n$ mutually exclusive events $E_{1}, \ldots, E_{n}$, and expect that any one of these, say $E_{i}$, indeed occurs “with probability” $p_{i}=\operatorname{Pr}\left(E_{i}\right)$. Then the parameters $p_{i}$ form a probability distribution $p \in \mathbb{R}^{\mathcal{E}}$ on the set $\mathcal{E}=\left{E_{1}, \ldots, E_{n}\right}$, i.e., the $p_{i}$ are nonnegative real numbers that sum up to 1 :
$$p_{1}+\cdots+p_{n}=1 \quad \text { and } \quad p_{1}, \ldots, p_{n} \geq 0$$
If we have furthermore a measuring or observation device $f$ that produces the number $f_{i}$ if $E_{i}$ occurs, then these numbers have the expected value
$$\mu(f)=f_{1} p_{1}+\cdots+f_{n} p_{n}=\sum_{k=1}^{n} f_{i} p_{i}=\langle f \mid p\rangle .$$
In a game-theoretic context, a probability is often a subjective evaluation of the likelihood for an event to occur. The gambler,investor, or general player may not know in advance what the future will bring, but has more or less educated guesses on the likelihood of certain events. There is a close connection with the notion of information.

Intensity. We think of the intensity of an event $E$ as a numerical parameter that is inversely proportional to its probability $p=\operatorname{Pr}(E)$ with which we expect its occurrence to be: the smaller $p$, the more intensely felt is an actual occurrence of $E$. For simplicity, let us take $1 / p$ as our objective intensity measure.

## 经济代写|博弈论代写Game Theory代考|Numbers and algebra

$$\mathbb{N}=1,2, \ldots, n, \ldots$$

$$\mathbb{N}_{0}=\mathbb{N} \cup 0$$

$$\mathrm{i}^{2}=-1$$

$$\mathbb{C}=a+\mathrm{i} b \mid a, b \in \mathbb{R}$$

$$\mathrm{i}=0+\mathrm{i} \cdot 1$$

## 经济代写|博弈论代写Game Theory代考|Probabilities, information and entropy

$$p_{1}+\cdots+p_{n}=1 \quad \text { and } \quad p_{1}, \ldots, p_{n} \geq 0$$

$$\mu(f)=f_{1} p_{1}+\cdots+f_{n} p_{n}=\sum_{k=1}^{n} f_{i} p_{i}=\langle f \mid p\rangle .$$

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## MATLAB代写

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