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

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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 Measures

The defining axioms of a probability measure as given in equations (2.22) through (2.25) correspond generally to intuitive notions, at least for the first three properties. The first property requires that a probability be a nonnegative number. In a purely mathematical sense, this is an arbitrary restriction, but it is in accord with the long history of intuitive and combinatorial developments of probability. Probability measures share this property with other measures such as area, volume, weight, and mass.
The second defining property corresponds to the notion that the probability that something will happen or that an experiment will produce one of its possible outcomes is one. This, too, is mathematically arbitrary but is a convenient and historical assumption. (From childhood we learn about things that are “100\% certain;” obviously we could as easily take 1 or $\pi$ (but not infinity – why?) to represent certainty.)

The third property, “additivity” or “finite additivity,” is the key one. In English it reads that the probability of occurrence of a finite collection of events having no points in common must be the sum of the probabilities of the separate events. More generally, the basic assumption of measure theory is that any measure – probabilistic or not – such as weight, volume, mass, and area should be additive: the mass of a group of disjoint regions of matter should be the sum of the separate masses; the weight of a group of objects should be the sum of the individual weights. Equation (2.24) only pins down this property for finite collections of events. The additional restriction of $(2.25)$, called countable additivity, is a limiting or asymptotic or infinite version, analogous to (2.19) for set algebra. This again leads to the rhetorical questions of why the more complicated, more restrictive, and less intuitive infinite version is required. In fact, it was the addition of this limiting property that provided the fundamental idea for Kolmogorov’s development of modern probability theory in the $1930 \mathrm{~s}$.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Limits of Probabilities

At times we are interested in finding the probability of the limit of a sequence of events. Th relate the conntahle additivity property of (9.25) to limiting properties, recall the discussion of the limiting properties of ovente given carlier in this chapter in terme of increaeing and decrene ing sequences of events. 引ay we have an increasing sequence of events $F_{n} ; n=0,1,2, \ldots, F_{n-1} \subset F_{n}$, and let $F$ denote the limit set, that is, the union of all of the $F_{n \text {. }}$. We have already argued that the limit set $F$ is itself an event. Intuitively, since the $F_{n}$ converge to $F$, the probabilities of the $F_{n}$ should converge to the probability of $F$. Such convergence is called a continuity property of probability and is very useful for evaluating the probabilities of complicated events as the limit of a sequence of probabili-

ties of simpler events. We shall show that countable additivity implies such continuity. To accomplish this, define the sequence of sets $G_{0}=F_{0}$ and $G_{n}=F_{n}-F_{n-1}$ for $n=1,2, \ldots$. The $G_{n}$ are disjoint and have the same union as do the $F_{n}$ (see Figure $2.2$ (a) as a visual aid). Thus we have from countable additivity that
\begin{aligned} P\left(\lim {n \rightarrow \infty} F{n}\right) &=P\left(\bigcup_{k=0}^{\infty} F_{k}\right) \ &=P\left(\bigcup_{k=0}^{\infty} G_{k}\right) \ &=\sum_{k=0}^{\infty} P\left(G_{k}\right) \ &=\lim {n \rightarrow \infty} \sum{k=0}^{n} P\left(G_{k}\right) \end{aligned}
where the last step simply uses the definition of an infinite sum. Since $G_{n}=F_{n}-F_{n-1}$ and $F_{n-1} \subset F_{n}, P\left(G_{n}\right)=P\left(F_{n}\right)-P\left(F_{n-1}\right)$ and hence
\begin{aligned} \sum_{k=0}^{n} P\left(G_{k}\right) &=P\left(F_{0}\right)+\sum_{k=1}^{n}\left(P\left(F_{n}\right)-P\left(F_{n-1}\right)\right) \ &=P\left(F_{n}\right) \end{aligned}
an example of what is called a telescoping sum” where each term cancels the previous term and adds a new piece, i.e.,
\begin{aligned} P\left(F_{n}\right)=& P\left(F_{n}\right)-P\left(F_{n-1}\right) \ +& P\left(F_{n-1}\right)-P\left(F_{n-2}\right) \ +& P\left(F_{n-2}\right)-P\left(F_{n-3}\right) \ & \vdots \ +& P\left(F_{1}\right)-P\left(F_{0}\right) \ +& P\left(F_{0}\right) \end{aligned}

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

We now provide several examples of probability measures on our examples of sample spaces and sigma-fields and thereby give some complete examples of probability spaces.

The first example formalizes the description of a probability measure as a sum of a pmf as introduced in the introductory section.
[2.12] Let $\Omega$ be a finite set and let $\mathcal{F}$ be the power set of $\Omega$. Suppose that we have a function $p(\omega)$ that assigns a real number to each sample point $\omega$ in such a way that
$$p(\omega) \geq 0, \text { all } \omega \in \Omega$$
and
$$\sum_{\omega \in \bar{\Omega}} p(\omega)=1$$

Define the set function $P$ by
\begin{aligned} P(F) &=\sum_{\omega \in F} p(\omega) \ &=\sum_{\omega \in \Omega} 1_{F}(\omega) p(\omega), \text { all } F \in \mathcal{F} \end{aligned}
where $1_{F}(\omega)$ is the indicator function of the set $F, 1$ if $\omega \in F$ and 0 otherwise.

For simplicity we drop the $\omega \in \Omega$ underneath the sum; that is, when no range of summation is explicit, it should be assumed the sum is over all possible values. Thus we can abbreviate (2.32) to
$$P(F)=\sum 1_{F}(\omega) p(\omega), \text { all } F \in \mathcal{F}$$
$P$ is easily verified to be a probability measure: It obviously satisfies axioms $2.1$ and 2.2. It is finitely and countably additive from the properties of sums. In particular, given a sequence of disjoint events, only a finite number can be distinct (since the power set of a finite space has only a finite number of members). To be disjoint, the balance of the sequence must equal $\emptyset$. The probability of the union of these sets will be the finite sum of the $p(\omega)$ over the points in the union which equals the sum of the probabilities of the sets in the sequence. Example [2.1] is a special case of example [2.12], as is the coin flip example of the introductary section.
The summation (2.33) used to define probability measures for a discrete space is a special case of a more general weighted sum, which we pause to define and consider. Suppose that $g$ is a real-valued function defined on $\Omega$, i.e., $g: \Omega \rightarrow \Re$ assigns a real number $g(\omega)$ to every $\omega \in \Omega$. We could consider more general complex-valued functions, but for the moment it is simpler to stick to real valued functions. Also, we could consider subsets of $\Re$, but we leave it more generally at this time. Recall that in the introductory section we considered such a function to be an example of signal processing and called it a random variable. Given a pmf $p$, define the expectation of $g$ (with respect to $p$ ) as
$$E(g)=\sum g(\omega) p(\omega) .$$
With this definition (2.33) with $g(\omega)=1_{F}(\omega)$ yields
$$P(F)=E\left(1_{F}\right),$$
${ }^{2}$ This is not in fact the fundamental definition of expectation that will be introduced in chapter 4 , but it will be seen to be equivalent.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Limits of Probabilities

∑ķ=0n磷(Gķ)=磷(F0)+∑ķ=1n(磷(Fn)−磷(Fn−1)) =磷(Fn)

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

[2.12] 让Ω是一个有限集，让F成为的幂集Ω. 假设我们有一个函数p(ω)为每个样本点分配一个实数ω以这样的方式
p(ω)≥0, 全部 ω∈Ω

∑ω∈Ω¯p(ω)=1

2这实际上不是第 4 章中将介绍的期望的基本定义，但它会被视为等价的。

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