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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 统计代写|统计推断代写Statistical inference代考|The Case of an Uncountable Outcomes Set S

As a prelude to the discussion that follows, let us see why the previous strategy of assigning probabilities to each and every outcome in the case of an uncountable set, say $S=\mathbb{R}$, will not work. The reason is very simple: the outcomes set has so many elements that it is impossible to arrange them in a sequence and thus count them. Hence, any attempt to follow the procedure used in the countable outcomes set case will lead to insurmountable difficulties. Intuitively, we know that we cannot cover the real line point by point. The only way to overlay $\mathbb{R}$ or any of its uncountable subsets is to use a sequence of intervals of any one of the following forms:
$(a, b),[a, b],[a, b),(-\infty, a]$, where $a<b, a$ and $b$ real numbers.
We will see in the sequel that the most convenient form for such intervals is ${(-\infty, x]}$ for each $x \in \mathbb{R}$.

In view of the above discussion, any attempt to define a random variable using the definition of a discrete random variable
$X(.): S \rightarrow \mathbb{R}_X$, such that ${s: X(s)=x}:=X^{-1}(x) \in \mathfrak{I}$ for all $x \in \mathbb{R}$
is doomed to failure. We have just agreed that the only way we can overlay $\mathbb{R}$ is by using intervals not points. The half-infinite intervals (3.10) suggest the modification of the events ${s: X(s)=x}$ of (3.11) into events of the form ${s: X(s) \leq x}$.

A random variable relative to $\Im$ is a function $X(.): S \rightarrow \mathbb{R}$ that satisfies the restriction
$${s: X(s) \leq x}:=X^{-1}((-\infty, x]) \in \mathfrak{I} \text { for all } x \in \mathbb{R} \text {. }$$
Notice that the only difference between this definition and that of a discrete random variable comes in the form of the events used.
$${s: X(s)=x} \subset{s: X(s) \leq x},$$
Moreover, in view of the fact that the latter definition includes the former as a special case. From this definition we can see that the pre-image of the random variable $X(.)$ takes us from intervals $(-\infty, x], x \in \mathbb{R}$ back to the event space $\Im$. The set of all such intervals generates a $\sigma$-field on the real line known as the Borel field and denoted by $\mathcal{B}(\mathbb{R})$ :
$$\mathcal{B}(\mathbb{R})=\sigma((-\infty, x], x \in \mathbb{R})$$

## 统计代写|统计推断代写Statistical inference代考|The Concept of a Cumulative Distribution Function

Using the concept of a random variable $X(.)$, so far we have transformed the abstract probability space $(S, \Im, \mathbb{P}(.))$ into a less abstract space $\left(\mathbb{R}, \mathcal{B}(\mathbb{R}), P_X(.)\right)$. However, we have not reached our target yet, because $P_X(.):=\mathbb{P} X^{-1}(.)$ is still a set function. Admittedly it is a much easier set function, because it is defined on the real line, but a set function all the same. What we prefer is a numerical point-to-point function.

The way we transform the set function $P_X$ (.) into a numerical point-to-point function is by a clever stratagem. By viewing $P_X(.)$ as only a function of the end point of the interval $(-\infty, x]$, we define the cumulative distribution function (cdf)
$$F_X(.): \mathbb{R} \rightarrow[0,1], \text { defined by } F_X(x)=\mathbb{P}{s: X(s) \leq x}=P_X((-\infty, x]) .$$

The ploy leading to this trick began a few pages ago when we argued that even though we could use any one of the following intervals (see Galambos, 1995):
$(a, b),[a, b],[a, b),(-\infty, a]$, where $a<b, a \in \mathbb{R}$, and $b \in \mathbb{R}$
to generate the Borel field $\mathcal{B}(\mathbb{R})$, we chose intervals of the form $(-\infty, x], x \in \mathbb{R}$.
In view of this, we can think of the cdf as being defined via
$$\mathbb{P}{s: a<X(s) \leq b}=\mathbb{P}{s: X(s) \leq b}-\mathbb{P}{s: X(s) \leq a}=P_X((a, b])=F_X(b)-F_X(a),$$
and then assume that $F_X(-\infty)=0$.
The properties of the $\operatorname{cdf} F_X(x)$ in Table $3.3$, where $x \rightarrow x_0^{+}$reads “as $x$ tends to $x_0$ through values greater than $x_0$,” are determined by those of $(S, \Im, \mathbb{P}(.))$. In particular from axioms [A1]-[A3] of $\mathbb{P}(.)$ and the mathematical structure of the $\sigma$-fields $\Im$ and $\mathcal{B}(\mathbb{R})$; see Karr (1993). That is, $F_X(x)$ is a non-decreasing, right-continuous function such that $F_X(-\infty)=0$, and $F_X(\infty)=1$. Properties $\mathrm{F} 1$ and $\mathrm{F} 3$ need no further explanation, but $\mathrm{F} 2$ is not obvious. The right-continuity property of the cdf follows from the axiom of countable additivity [3] of the probability set function $\mathbb{P}(.)$, whose value stems from the fact that at every point of discontinuity $x_0, \mathrm{~F} 2$ holds.

# 统计推断代考

## 统计代写|统计推断代写统计推断代考|不可数结果集S的情况

$(a, b),[a, b],[a, b),(-\infty, a]$，其中$a<b, a$和$b$实数。我们将在续集中看到，这种间隔最方便的形式是${(-\infty, x]}$对应每个$x \in \mathbb{R}$ .

$X(.): S \rightarrow \mathbb{R}_X$定义随机变量的尝试，使${s: X(s)=x}:=X^{-1}(x) \in \mathfrak{I}$对于所有$x \in \mathbb{R}$

$${s: X(s) \leq x}:=X^{-1}((-\infty, x]) \in \mathfrak{I} \text { for all } x \in \mathbb{R} \text {. }$$

$${s: X(s)=x} \subset{s: X(s) \leq x},$$

$$\mathcal{B}(\mathbb{R})=\sigma((-\infty, x], x \in \mathbb{R})$$ 表示

## 统计代写|统计推断代写统计推断代考|累积分布函数的概念

$$F_X(.): \mathbb{R} \rightarrow[0,1], \text { defined by } F_X(x)=\mathbb{P}{s: X(s) \leq x}=P_X((-\infty, x]) .$$

$(a, b),[a, b],[a, b),(-\infty, a]$，其中$a<b, a \in \mathbb{R}$，和$b \in \mathbb{R}$

$$\mathbb{P}{s: a<X(s) \leq b}=\mathbb{P}{s: X(s) \leq b}-\mathbb{P}{s: X(s) \leq a}=P_X((a, b])=F_X(b)-F_X(a),$$
，然后假设$F_X(-\infty)=0$ .

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## MATLAB代写

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assignmentutor™您的专属作业导师
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