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# 统计代写|概率与统计代考Probability and Statistics代写|STA312 NECESSITY OF THE AXIOMS

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## 统计代写|概率与统计代考Probability and Statistics代写|NECESSITY OF THE AXIOMS

Looking at Axiom 3, one may wonder why do we need it for the case of countable (and not just finite) sequences of events. Indeed, the necessity of all three axioms, with only finite additivity in Axiom 3, can be easily justified simply by using probability to represent the limiting relative frequency of occurrences of events. Recall the symbol $N(A)$ from Section $2.1$ for the number of occurrences of the event $A$ in the first $N$ experiments. The nonnegativity axiom is simply a reflection of the fact that the count $N(A)$ cannot be negative. The norming axiom reflects the fact that event $\mathcal{S}$ is certain and must occur in every experiment so that $N(\mathcal{S})=N$, and hence, $N(\mathcal{S}) / N=1$. Finally, (taking the case of two disjoint events $A$ and $B$ ), we have $N(A \cup B)=N(A)+N(B)$, since whenever $A$ occurs, $B$ does not, and conversely. Thus, if probability is to reflect the limiting relative frequency, then $P(A \cup B)$ should be equal to $P(A)+P(B)$, since the frequencies satisfy the analogous condition $N(A \cup B) / N=N(A) / N+N(B) / N$.

The need for countable additivity, however, cannot be explained so simply. This need is related to the fact that to build a sufficiently powerful theory, one needs to take limits. If $A_1, A_2, \ldots$ is a monotone sequence of events (increasing or decreasing, i.e., $A_1 \subset A_2 \subset \cdots$ or $\left.A_1 \supset A_2 \cdots\right)$ then $\lim P\left(A_n\right)=P\left(\lim A_n\right)$, where the event $\lim A_n$ has been defined in Section 1.4. Upon a little reflection, one can see that such continuity is a very natural requirement. In fact, the same requirement has been taken for granted for over 2,000 years in a somewhat different context: in computing the area of a circle, one uses a sequence of polygons with an increasing number of sides, all inscribed in the circle. This leads to an increasing sequence of sets “converging” to the circle, and therefore the area of the circle is taken to be the limit of the areas of approximating polygons. The validity of this idea (i.e., the assumption of the continuity of the function $f(A)=$ area of $A$ ) was not really questioned until the beginning of the twentieth century. Research on the subject culminated with the results of Lebesgue.

## 统计代写|概率与统计代考Probability and Statistics代写|SUBJECTIVE PROBABILITY

Let us finally consider briefly the third interpretation of probability, namely as a degree of certainty, or belief, about the occurrence of an event. Most often, this probability is associated not so much with an event as with the truth of a proposition asserting the occurrence of this event.

The material of this section assumes some degree of familiarity with the concept of expectation, formally defined only in later chapters. For the sake of completeness, in the simple form needed here, this concept is defined below. In the presentation, we follow more or less the historical development, refining gradually the conceptual structures introduced. The basic concept here is that of a lottery, defined by an event, say $A$, and two objects, say $a$ and $b$. Such a lottery, written simply $a A b$, will mean that the participant $(\mathrm{X})$ in the lottery receives object $a$ if the event $A$ occurs, and receives object $b$ if the event $A^c$ occurs.
The second concept is that of expectation associated with the lottery $a A b$, defined as
$$u(a) P(A)+u(b) P\left(A^c\right),$$
where $u(a)$ and $u(b)$ are measures of how much the objects $a$ and $b$ are “worth” to the participant. When $a$ and $b$ are sums of money (or prices of objects $a$ and $b$ ), and we put $u(x)=x$, the quantity (2.13) is sometimes called expected value. In cases where $u(a)$ and $u(b)$ are values that person $\mathrm{X}$ attaches to $a$ and $b$ (at a given moment), these values do not necessarily coincide with prices. We then refer to $u(a)$ and $u(b)$ as utilities of $a$ and $b$, and the quantity (2.13) is called expected utility $(E U)$. Finally, when in the latter case, the probability $P(A)$ is the subjective assessment of likelihood of the event $A$ by $\mathrm{X}$, the quantity (2.13) is called subjective expected utility $(S E U)$.

First, it has been shown by Ramsey (1926) that the degree of certainty about the occurrence of an event (of a given person) can be measured. Consider an event $A$, and the following choice suggested to $\mathrm{X}$ (whose subjective probability we want to determine). $\mathrm{X}$ is namely given a choice between the following two options:

1. Sure option: receive some fixed amount $\$ u$, which is the same as lottery$(\$u) B(\$ u)$, for any event$B$. 2. A lottery option. Receive some fixed amount, say$\$100$, if $A$ occurs, and receive nothing if $A$ does not occur, which is lottery $(\$ 100) A(\$0)$. One should expect that if $u$ is very small, $\mathrm{X}$ will probably prefer the lottery. On the other hand, if $u$ is close to $\$ 100, \mathrm{X}$may prefer the sure option. Therefore, there should exist an amount$u^$such that$\mathrm{X}$will be indifferent between the sure option with$u^$and the lottery option. With the amount of money as a representation of its value (or utility), the expected return from the lottery equals $$0(1-P(A))+100 P(A)=100 P(A),$$ which, in turn, equals$u^$. Consequently, we have$P(A)=u^ / 100$. Obviously, under the stated assumption that utility of money is proportional to the dollar amount, the choice of$\$100$ is not relevant here, and the same value for $P(A)$ would be obtained if we choose another “base value” in the lottery option (this can be tested empirically).

# 概率与统计代写

## 统计代写|概率与统计代考概率与统计代写|主观概率

$$u(a) P(A)+u(b) P\left(A^c\right),$$
，其中$u(a)$和$u(b)$是衡量对象$a$和$b$对参与者的“价值”有多少。当$a$和$b$是钱的总和(或物品的价格$a$和$b$)，我们写上$u(x)=x$，数量(2.13)有时被称为期望值。如果$u(a)$和$u(b)$是$\mathrm{X}$附加在$a$和$b$上的值(在给定时刻)，这些值不一定与价格一致。然后我们将$u(a)$和$u(b)$称为$a$和$b$的效用，数量(2.13)称为期望效用$(E U)$。最后，当在后一种情况下，概率$P(A)$是$\mathrm{X}$对事件$A$的可能性的主观评价时，这个量(2.13)称为主观期望效用$(S E U)$。

1. 确定选项:接收一定的固定金额$\$ u$，相当于彩票$(\$u) B(\$ u)$，任何事件$B$2. 抽奖选项。如果$A$出现，就收到固定的金额，比如$\$100$，如果$A$没有出现，就什么也得不到，这就是彩票$(\$ 100) A(\$0)$。人们应该预料到，如果$u$非常小，$\mathrm{X}$可能更喜欢抽签。另一方面，如果$u$接近$\$ 100, \mathrm{X}$可能更喜欢确定的选项。 因此，应该存在一个数量$u^$，使得$\mathrm{X}$在带有$u^$的确定选项和抽签选项之间是无所谓的。用钱的数量来表示它的价值(或效用)，彩票的预期收益等于 $$0(1-P(A))+100 P(A)=100 P(A),$$ ，反过来等于$u^$。因此，我们有$P(A)=u^ / 100$。显然，在假定货币效用与美元金额成正比的前提下，选择$\$100$在这里是无关的，如果我们在抽签选项中选择另一个“基础值”(这可以通过经验检验)，将获得相同的$P(A)$值。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。