### 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Probability Spaces

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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

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Probability Spaces

We now turn to a more thorough development of the ideas introduced in the previous section.

A sample space $\Omega$ is an abstract space, a nonempty collection of points or members or elements called sample points (or elementary events or elementary outcomes).

An event space (or sigma-field or sigma-algebra) $\mathcal{F}$ of a sample space $\Omega$ is a nonempty collection of stabsets of $\Omega$ called events with the following properties:
If $F \in \mathcal{F}$, then also $F^{e} \in \mathcal{F}$,
that is, if a given set is an event, then its complement must also be an event. Note that any particular subset of $\Omega$ may or may not be an event (review the quantizer example).
If for some finite $n, F_{i} \in \mathcal{F}, i=1,2, \ldots, n$, then also
$$\bigcup_{i=1}^{n} F_{i} \in \mathcal{F}$$

that is, a finite union of events must also be an event.
If $F_{i} \in \mathcal{F}, i=1,2, \ldots$, then also
$$\bigcup_{i=1}^{\infty} F_{i} \in \mathcal{F} \text {, }$$
that is, a countable union of events must also be an event.
We shall later see alternative ways of describing (2.19), but this form is the most common.

Eq. (2.18) can be considered as a special case of (2.19) since, for example, given a finite collection $F_{i} ; i=1, \ldots, N$, we can construct an infinite sequence of sets with the same union, e.g., given $F_{k}, k=1,2, \ldots, N$, construct an infinite sequence $G_{n}$ with the same union by choosing $G_{n}=F_{n}$ for $n=1,2, \ldots N$ and $G_{n}=\emptyset$ otherwise. It is convenient, however, to consider the finite case separately. If a collection of sets satisfies only (2.17) and (2.18) but not $2.19$, then it is called a field or algebra of sets. For this reason, in elementary probability theory one often refers to “set algebra” or to the “algebra of events.” (Don’t worry about why $2.19$ might not be satisfied.) Both (2.17) and (2.18) can be considered as “closure” properties; that is, an event space must be closed under complementation and unions in the sense that performing a sequence of complementations or unions of events must yield a set that is also in the collection, i.e., a set that is also an event. Observe also that (2.17), (2.18), and (A.11) imply that
$$\Omega \in \mathcal{F} .$$
that is, the whole sample space considered as a set must be in $\mathcal{F}$; that is, it must be an event. Intuitively, $\Omega$ is the “certain event,” the event that “something happens.” Similarly, (2.20) and (2.17) imply that
$$\theta \in \mathcal{F} \text {, }$$
and hence the empty set must be in $\mathcal{F}$, corresponding to the intuitive event “nothing happens.”

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Sample Spaces

Intuitively, a sample space is a listing of all conceivable finest-grain, distin$\mathrm{~ g u i s h a ̆ b l e ́ ~ o u t c o o m e ̀ s ~ o ́ ~ a n ~ e x p e r r i m e ̀ n t ~ t o ́ ~ b e ́ ~ m o ́ d e ̀ l e ́ d ~ b y ~ a ́ ~ p r o b b a}$ Mathematically it is just au abstrast spase.
Examples
[2.2] A finite space $\Omega=\left{a_{k} ; k=1,2, \ldots, K\right}$. Specific examples are the binary space ${0,1}$ and the finite space of integers $\mathcal{Z}_{k} \triangleq{0,1,2, \ldots, k-$ 1].

[2.3] A countably infinite space $\Omega=\left{a_{k} ; k=0,1,2, \ldots\right}$, for some sequence $\left{a_{k}\right}$. Specific examples are the space of all nonnegative integers ${0,1,2, \ldots}$, which we denote by $\mathcal{Z}_{+}$, and the space of all integers ${\ldots,-2,-1,0,1,2, \ldots}$, which we denote by $\mathcal{Z}$. Other examples are the space of all rational numbers, the space of all even integers, and the space of all periodic sequences of integers.

Both examples [2.2] and [2.3] are called discrete spaces. Spaces with finite or countably infinite numbers of elements are called discrete spaces.
[2.4] An interval of the real line $\Re$, for example, $\Omega=(a, b)$. We might consider an open interval $(a, b)$, a closed interval $[a, b]$, a half-open interval $[a, b)$ or $(a, b]$, or even the entire real line ${$ itself. (See appendix $\mathrm{A}$ for details on these different types of intervals.)

Spaces such as example [2.4] that are not discrete are said to be continuous. In some cases it is more accurate to think of spaces as being a mixture of discrete and continuous parts, e.g., the space $\Omega=(1,2) \cup{4}$ consisting of a continuous interval and an isolated point. Such spaces can usually be handled by treating the discrete and continuous components separately.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Event Spaces

Intuitively, an event space is a collection of subsets of the sample space or groupings of elementary events which we shall consider as physical events and to which we wish to assign probabilities. Mathematically, an event space is a collection of subsets that is closed under certain set-theoretic operations; that is, performing certain operations on events or members of the event space must give other events. Thus, for example, if in the example of a single voltage measurement we have $\Omega=\Re$ and we are told that the set of all voltages greater than 5 volts $={\omega: \omega \geq 5}$ is an event, that is, is a member of a sigma-field $\mathcal{F}$ of subsets of $\mathcal{F}$, then necessarily its complement ${\omega: \omega<5}$ must also be an event, that is, a member of the sigma-field $\mathcal{F}$. If the latter set is not in $\mathcal{F}$ then $\mathcal{F}$ cannot be an event space! Observe that no problem arises if the complement physically cannot happen – events that “cannot occur” can be included in $F$ and then assigned probability zero when choosing the probability measure $P$. For example, even if you know that the voltage does not exceed 5 volts, if you have chosen the real line $x$ as your sample space, then you must include the set ${r: r>5}$ in the event space if the set ${r: r \leq 5}$ is an event. The impossibility of a voltage greater than 5 is then expressed by assigning $P({r: r>5})=0$.
While the definition of a sigma-field requires only that the class be closed under complementation and countable unions, these requirements immediately yield additional closure properties. The countably infinite version of DeMorgan’s “laws” of elementary set theory require that if $F_{i}, i=1,2, \ldots$ are all members of a sigma-field, then so is
$$\bigcap_{i=1}^{\infty} F_{i}=\left(\bigcup_{i=1}^{\infty} F_{i}^{c}\right)^{e} \text {. }$$
It follows by similar set-theoretic arguments that any countable sequence of any of the set-theoretic operations (union, intersection, complementation, difference, symmetric difference) performed on events must yield other events. Ohserve, however, that there is no guarantee that uncountabte operations on events will produce new events; they may or may not. For example, if we are told that $\left{F_{r} ; r \in[0,1]\right}$ is a family of events, then it is not necessarily true that $\bigcup_{r \in[0,1]} F_{r}$, is an event (see problem $2.2$ for an example).

⋃一世=1nF一世∈F

⋃一世=1∞F一世∈F,

Ω∈F.

θ∈F,

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Sample Spaces

[2.2] 有限空间\Omega=\left{a_{k} ; k=1,2, \ldots, K\right}\Omega=\left{a_{k} ; k=1,2, \ldots, K\right}. 具体例子是二进制空间0,1和整数 $\mathcal{Z}_{k} \triangleq{0,1,2, \ldots, k-$ 1] 的有限空间。

[2.3] 可数无限空间\Omega=\left{a_{k} ; k=0,1,2, \ldots\right}\Omega=\left{a_{k} ; k=0,1,2, \ldots\right}, 对于某个序列\left{a_{k}\right}\left{a_{k}\right}. 具体例子是所有非负整数的空间0,1,2,…，我们表示为从+, 和所有整数的空间…,−2,−1,0,1,2,…，我们表示为从. 其他例子是所有有理数的空间，所有偶数的空间，以及所有周期整数序列的空间。

[2.4] 实线区间ℜ， 例如，Ω=(一种,b). 我们可以考虑开区间(一种,b), 闭区间[一种,b], 半开区间[一种,b)或者(一种,b]，甚至整条实线 ${一世吨s和lF.(小号和和一种pp和nd一世X\mathrm{A}$ 了解这些不同类型区间的详细信息。）

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Event Spaces

⋂一世=1∞F一世=(⋃一世=1∞F一世C)和.

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