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

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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|Random Processes

It is straightforward conceptually to go from one random variable to $k$ random variables constituting a $k$-dimensional random vector. It is perhaps a greater leap to extend the idea to a random process. The idea is at least easy to state, but it will take more work to provide examples and the mathematical details will prove more complicated. A random process is a sequence of random variables $\left{X_{n} ; n=0,1, \ldots\right}$ defined on a common experiment. It can be thought of as an infinite dimensional random vector. To be more accurate, this is an example of a discrete-time, one-sided random process. It is called “discrete-time” because the index $n$ which corresponds to time takes on discrete values (here the nonnegative integers) and it is called “one-sided” because only nonnegative times are allowed. A discrete-time random process is also called a time series in the statistics literature and it is often denoted as ${X(n) n=0,1, \ldots}$ and is sometimes denoted by ${X[n]}$ in the digital signal processing literature. Two questions might oocur to the reader: how does one construct an infinite family of random variables on a single experiment? How can one provide a direct development of a random process as accomplished for random variables and vectors? The direct development might appear hopeless since infinite dimensional vectors are involved.

The first problem is reasonably easy to handle by example. Consider the usual uniform pdf experiment. Rename the random variables $Y$ and $W$ as $X_{0}$ and $X_{1}$, respectively. Consider the following definition of an infinite family of random variables $X_{n}:[0,1) \rightarrow{0,1}$ for $n=0,1, \ldots$. Every $r \in[0,1)$ can be expanded as a binary expansion of the form
$$r=\sum_{n=0}^{\infty} b_{n}(r) 2^{-n-1}$$
This simply replaces the usual decimal representation by a binary representation. For example, $1 / 4$ is $.25$ in decimal and .01 or .010000… in binary. $1 / 2$ is .5 in decimal and yields the binary sequence .1000…, $1 / 4$ is $.25$ in decimal and yields the binary sequence .0100…, $3 / 4$ is .75 in decimal and $.11000 \ldots .$ and $1 / 3$ is $.3333 . . .$ in decimal and $.010101 \ldots$ in binary.

Define the random process by $X_{n}(r)=b_{n}(r)$, that is, the $n$th term in the binary expansion of $r$. When $n=0,1$ this reduces to the specific $X_{0}$ and $X_{1}$ already considered.

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

We now develop the promised precise definition of a random variable. As you might guess, a technical condition for random variables is required because of certain subtle pathological problems that have to do with the ability to determine probabilities for the random variable. To arrive at the precise definition, we start with the informal definition of a random variable that we have already given and then show the inevitable difficulty that results without the technical condition. We have informally defined a

random variable as being a function on a sample space. Suppose we have a probability space $(\Omega, \mathcal{F}, P)$. Let $f: \Omega \rightarrow \Re$ be a function mapping the same space into the real line so that $f$ is a candidate for a random variable. Since the selection of the original sample point $\omega$ is random, that is, governed by a probability measure, so should be the output of our measurement of random variable $f(\omega)$. That is, we should be able to find the probability of an “output event” such as the event “the outcome of the random variable $f$ was between $a$ and $b, “$ that is, the event $F \subset \Re$ given by $F=(a, b)$. Observe that there are two different kinds of events being considered here:

1. output events or members of the event space of the range or range space of the random variable, that is, events consisting of subsets of possible output values of the random variable; and
2. input events or $\Omega$ events, events in the original sample space of the original probability space.

Can we find the probability of this output event? That is, can we make mathematieal senee out of the quantity “the probability that $f$ areumee a value in an event $F \subset \Re$ ?? On reflection it seems clear that we can. The probability that $f$ assumes a value in some set of values must be the probability of all values in the original sample space that result in a value of $f$ in the given set. We will make this concept more precise shortly. To save writing we will abbreviate such English statements to the form $\operatorname{Pr}(f \in F)$, or $\operatorname{Pr}(F)$, that is, when the notation $\operatorname{Pr}(F)$ is encountered it should be interpreted as shorthand for the English statement for “the probability of an event $F^{” \prime}$ or “the probability that the event $F$ will occur” and not as a precise mathematical quantity.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Distributions of Random Variables

Suppose we have a probability space $(\Omega, \mathcal{F}, P)$ with a random variable, $X$, defined on the space. The random variable $X$ takes values on its range space which is some subset $A$ of $\Re$ (possibly $A=\Re$ ). The range space $A$ of a random variable is often called the alphabet of the random variable. As we have seen, since $X$ is a random variable, we know that all subsets of $\Omega$ of the form $X^{-1}(F)={\omega: X(\omega) \in F}$, with $F \in B(A)$, must be members of $\mathcal{F}$ by definition. Thus the set function $P_{X}$ defined by
$$P_{X}(F)=P\left(X^{-1}(F)\right)=P({\omega: X(\omega) \in F}) ; F \in \mathcal{B}(A)$$
is well defined and assigns probabilities to output events involving the random variable in terms of the original probability of input events in the orig-

inal experiment. The three written forms in equation (3.22) are all read as $\operatorname{Pr}(X \in F)$ or “the probability that the random variable $X$ takes on a value in $F .$ Furthermore, since inverse images preserve all set-theoretic operations (see problem A.12), $P_{X}$ satisfies the axioms of probability as a probability measure on $(A, \mathcal{B}(A))-$ it is nonnegative, $P_{X}(A)=1$, and it is countably additive. Thus $P_{X}$ is a probability measure on the measurable space $(A, \mathcal{B}(A))$. Therefore, given a probability space and a random variable $X$, we have constructed a new probability space $\left(A, \mathcal{B}(A), P_{X}\right)$ where the events describe outcomes of the random variable. The probability measure $P_{X}$ is called the distribution of $X$ (as opposed to a “cumulative distribution function” of $X$ to be introduced later).

If two random variables have the same distribution, then they are said to be equivalent since they have the same probabilistic description, whether or not they are defined on the same underlying space or have the same functional form (see problem 3.22).

A substantial part of the application of probability theory to practical probblems is devoted to determining the distributions of random variables, perfurmaing the “ealeulus of prubabsility.” Ons bugina with a probubility space. A random variable is defined on that space. The distribution of the random variable is then derived, and this results in a new probability space. This topic is called variously “derived distributions” or “transformations of random variables” and is often developed in the literature as a sequence of apparently unrelated subjects. When the points in the original sample space can be interpreted as “signals,” then such problems can be viewed as “signal processing” and derived distribution problems are fundamental to the analysis of statistical signal processing systems. We shall emphasize that all such examples are just applications of the basic inverse image formula (3.22) and form a unified whole. In fact, this formula, with its vector analog, is one of the most important in applications of probability theory. Its specialization to discrete input spaces using sums and to continuous input spaces using integrals will be seen and used often throughout this book.

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

r=∑n=0∞bn(r)2−n−1

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

1. 输出事件或随机变量的范围或范围空间的事件空间的成员，即由随机变量的可能输出值的子集组成的事件；和
2. 输入事件或Ω事件，原始概率空间的原始样本空间中的事件。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Random Variables, Vectors, and Processes

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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|Random Variables

The name random variable suggests a variable that takes on values randomly. In a loose, intuitive way this is the right interpretation – e.g., an observer who is measuring the amount of noise on a communication link sees a random variable in this sense. We require, however, a more precise mathematical definition for analytical purposes. Mathematically a random variable is neither random nor a variable – it is just a function mapping one sample space into another space. The first space is the sample space portion of a probability space, and the second space is a subset of the real line (some authors wotld call this a “real-valued” random variable). The careful mathematical definition will place a constraint on the function to ensure that the theory makes sense, but for the moment we will adopt the informal definition that a random variable is just a function.

A random variable is perhaps best thought of as a measurement on a

probability space; that is, for each sample point $\omega$ the random variable produces some value, denoted functionally as $f(\omega)$. One can view $\omega$ as the result of some experiment and $f(\omega)$ as the restalt of a measurement made on the experiment, as in the example of the simple binary quantizer introduced in the introduction to chapter 2 . The experiment outcome $\omega$ is from an abstract space, e.g., real numbers, integers, ASCII characters, waveforms, sequences, Chinese characters, etc. The resulting value of the measurement or random variable $f(\omega)$, however, must be “concrete” in the sense of being a real number, e.g., a meter reading. The randomness is all in the original probability space and not in the random variable; that is, once the $\omega$ is selected in a “random” way, the output value of sample value of the random variable is determined.

Alternatively, the original point $\omega$ can be viewed as an “input signal” and the random variable $f$ can be viewed as “signal processing,” i.e., the input signal $\omega$ is converted into an “output signal” $f(\omega)$ by the random variable. This viewpoint becomes both precise and relevant when we indeed choose our original sample space to be a signal space and we generalize random variables by random vectors and processes.

Before proceeding to the formal definition of random variables, vectors, and processes, we motivate several of the basic ideas by simple examples, beginning with random variables constructed on the fair wheel experiment of the introduction to chapter 2 .

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|A Coin Flip

We have already encountered an example of a random variable in the introduction to chapter 2 , where we defined a random variable $q$ on the spinning wheel experiment which produced an output with the same pmf as a uniform coin flip. We begin by summarizing the idea with some slight notational changes and then consider the implications in additional detail.
Begin with a probability space $(\Omega, \mathcal{F}, P)$ where $\Omega=\Re$ and the probability $P$ is defined by (2.2) using the uniform pdf on $[0,1)$ of (2.4) Define the function $Y: \Re \rightarrow{0,1}$ by
$$Y(r)= \begin{cases}0 & \text { if } r \leq 0.5 \ 1 & \text { otherwise }\end{cases}$$
When Tyche performs the experiment of spinning the pointer, we do not actually observe the pointer, but only the resulting binary value of $Y$. $Y$ can be thought of as signal processing or as a measurement on the original experiment. Subject to a technical constraint to be introduced later, any function defined on the sample space of an experiment is called a random

variable. The “randomness” of a random variable is “inherited” from the underlying experiment and in theory the probability measure describing its outputs should be derivable from the initial probability space and the structure of the function. To avoid confusion with the probability measure $P$ of the original experiment, refer to the probability measure associated with outcomes of $Y$ as $P_{Y} . P_{Y}$ is called the distribution of the random variable $Y$. The probability $P_{Y}(F)$ can be defined in a natural way as the probability computed using $P$ of all the original samples that are mapped by $Y$ into the subset $F$ :
$$P_{Y}(F)=P({r: Y(r) \in F})$$

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

The issue of the possible equality of two random variables raises an interesting point. If you are told that $Y$ and $V$ are two separate random variables with pm’t’s $p_{Y}$ and $p_{V}$, then the quegtion of whether or not they are equivalent can be answered from these pmf’s alone. If you wish to determine whether or not the two random variables are in fact equal, however, then they must be considered together or jointly. In the case where we have a random variable $Y$ with outcomes in ${0,1}$ and a random variable $V$ with outcomes in ${0,1}$, we could consider the two together as a single random vector ${Y, V}$ with outcomes in the Cartesian product space $\Omega_{Y V}={0,1}^{2} \triangleq{(0,0),(0,1),(1,0),(1,1)}$ with some pmf $p_{Y, V}$ describing the combined behavior
$$p_{Y, V}(y, v)=\operatorname{Pr}(Y=y, V=v)$$
so that
$$\operatorname{Pr}((Y, V) \in F)=\sum_{y, v:(y, v) \in F} p_{Y, V}(y, v) ; F \in \mathcal{B}{Y V},$$ where in this simple discrete problem we take the event space $\mathcal{B}{Y V}$ to be the power set of $\Omega_{Y V}$. Now the question of equality makes sense as we can evaluate the probability that the two are equal:
$$\operatorname{Pr}(Y=V)=\sum_{y, v \in Y-v} p_{Y, V}(y, v) .$$
If this probability is 1 , then we know that the two random variables are in fact equal with probability $1 .$

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

p是,在(是,在)=公关⁡(是=是,在=在)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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|Independence

Given a probability space $(\Omega, F, P)$, two events $F$ and $G$ are defined to be independent if $P(F \cap G)=P(F) P(G)$. A collection of events $\left{F_{i} ; i=\right.$ $0,1, \ldots, k-1}$ is said to be independent or mutually independent if for any distinct subcollection $\left{F_{I_{i}} ; i=0,1, \ldots, m-1\right}, l_{m} \leq k$, we have that
$$P\left(\prod_{i=0}^{m-1} F_{l_{i}}\right)=\prod_{i=0}^{m-1} P\left(F_{l_{i}}\right) .$$
In words: the probability of the intersection of any subcollection of the given events equals the product of the probabilities of the separate events. Unfortunately it is not enough to simply require that $P\left(\bigcap_{i=0}^{k-1} F_{i}\right)=\prod_{i=0}^{k-1} P\left(F_{i}\right)$

as this does not imply a similar result for all possible subcollections of events, which is what will be needed. For example, consider the following case where $P(F \cap G \cap H)=P(F) P(G) P(H)$ for three events $F$, $G$, and $H$, yet it is not true that $P(F \cap G)=P(F) P(G)$
\begin{aligned} P(F) &=P(G)=P(H)=\frac{1}{3} \ P(F \cap G \cap H) &=\frac{1}{27}=P(F) P(G) P(H) \ P(F \cap G) &=P(G \cap H)=P(F \cap H)=\frac{1}{27} \neq P(F) P(G) . \end{aligned}
The example places zero probability on the overlap $F \cap G$ except where it also overlaps $H$, i.e., $P\left(F \cap G \cap H^{c}\right)=0$. Thus in this case $P(F \cap G \cap H)=$ $P(F) P(G) P(H)=1 / 27$, but $P(F \cap G)=1 / 27 \neq P(F) P(G)=1 / 9$.

The concept of independence in the probabilistic sense we have defined relates easily to the intuitive idea of independence of physical events. For example, if a fair die is rolled twice, one would expect the second roll to be unrelated to the first roll because there is no physical connection between the individual outcomes. Independence in the probabilistic sense is reflected in this experiment. The probability of any given outcome for either of the individual rolls is $1 / 6$. The probability of any given pair of outcomes is $(1 / 6)^{2}=1 / 36$ – the addition of a second outcome diminishes the overall probability by exactly the probability of the individual event, viz., $1 / 6$. Note that the probabilities are not added – the probability of two successive outcomes cannot reasonably be greater than the probability of either of the outcomes alone. Do not, however, confuse the concept of independence with the concept of disjoint or mutually exclusive events. If you roll the die once, the event the roll is a one is not independent of the event the roll is a six. Given one event, the other cannot happen they are neither physically nor probabilistically independent. These are mutually exclusive events.

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

Intuitively, independence of two events means that the occurrence of one event should not affect the occurrence of the other. For example, the knowledge of the outcome of the first roll of a die should not change the probabilities for the outcome of the second roll of the die if the die has no memory. To be more precise, the notion of conditional probability is required. Consider the following motivation. Suppose that $(\Omega, \mathcal{F}, P)$ is a probability space and that an observer is told that an event $G$ has already occurred. The

observer thus has a posteriori knowledge of the experiment. The observer is then asked to calculate the probability of another event $F$ given this information. We will denote this probability of $F$ given $G$ by $P(F \mid G)$. Thus instead of the a priori or unconditional probability $P(F)$, the observer must compute the a posteriori or conditional probability $P(F \mid G)$, read as “the probability that event $F$ occurs given that the event $G$ occurred.” For a fixed $G$ the observer should be able to find $P(F \mid G)$ for all events $F$, thus the observer is in fact being asked to describe a new probability measure, say $P_{G}$, on $(\Omega, \mathcal{F})$. How should this be defined? Intuition will lead to a useful definition and this definition will indeed provide a useful interpretation of independence.

First, since the observer has been told that $G$ has occurred and hence $\omega \in G$, clearly the new probability measure $P_{G}$ must assign zero probability to the set of all $\omega$ outside of $G$, that is, we should have
$$P\left(G^{c} \mid G\right)=0$$
or, equivalently,
$$P(G \mid G)=1 .$$
Eq. (2.91) plus the axioms of probability in turn imply that
$$P(F \mid G)=P\left(F \cap\left(G \cup G^{c}\right) \mid G\right)=P(F \cap G \mid G) .$$
Second, there is no reason to suspect that the relative probabilities within $G$ should change because of the conditioning. For example, if an event $F \subset G$ is twice as probable as an event $H \subset G$ with respect to $P$, then the same should be true with respect to $P_{G}$. For arbitrary events $F$ and $H$, the events $F \cap G$ and $H \cap G$ are both in $G$, and hence this preservation of relative probability implies that
$$\frac{P(F \cap G \mid G)}{P(H \cap G \mid G)}=\frac{P(F \cap G)}{P(H \cap G)} .$$
But if we take $H=\Omega$ in this formula and use (2.92)-(2.93), we have that
$$P(F \mid G)=P(F \cap G \mid G)=\frac{P(F \cap G)}{P(G)},$$
which is in fact the formula we now use to define the conditional probability of the event $F$ given the event $G$. The conditional probability can be interpreted as “cutting down” the original probability space to a probability space with the smaller sample space $G$ and with probabilities equal to the renormalized probabilities of the intersection of events with the given event $G$ on the original space.

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

1. Suppose that you have a set function $P$ defined for all subsets $F \subset \Omega$ of a sample space $\Omega$ and suppose that you know that this set function satisfies (2.7-2.9). Show that for arbitrary (not necessarily disjoint) events.
$$P(F \cup G)=P(F)+P(G)-P(F \cap G) .$$
2. Describe the sigma-field of subsets of $\Re$ generated by the points or singleton sets. Does this sigma-field contain intervals of the form $(a, b)$ for $b>a$ ?
3. Given a finite subset $A$ of the real line $\Re$, prove that the power set of $A$ and $\mathcal{B}(A)$ are the same. Repeat for a countably infinite subset of $\Re$.
4. Given that the discrete sample space $\Omega$ has $n$ elements, show that the power set of $\Omega$ consists of $2^{n}$ elements.
5. ${ }^{*}$ Let $\Omega=\Re$, the real line, and consider the collection $\mathcal{F}$ of subsets of $\Re$ defined as all sets of the form
$$\bigcup_{i=0}^{k}\left(a_{i}, b_{i}\right] \cup \bigcup_{j=0}^{m}\left(c_{j}, d_{j}\right]^{e}$$
for all possible choices of nonnegative integers $k$ and $m$ and all possible choices of real numbers $a_{i}<b_{i}, c_{i}<d_{i}$. If $k$ or $m$ is 0 , then the respective unions are defined to be empty so that the empty set itself has the form given. In other words, $\mathcal{F}$ contains all possible finite unions of half-open intervals of this form and complements of such half-open intervals. Every set of this form is in $\mathcal{F}$ and every set in $\mathcal{F}$ has this form. Prove that $\mathcal{F}$ is a field of subsets of $\Omega$. Does $\mathcal{F}$ contain the points? For example, is the singleton set ${0}$ in $\mathcal{F}$ ? Is $\mathcal{F}$ a sigma-field?
6. Let $\Omega=[0, \infty)$ be a sample space and let $\mathcal{F}$ be the sigma-field of subsets of $\Omega$ generated by all sets of the form $(n, n+1)$ for $n=1,2, \ldots$

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

1. 假设你有一个集合函数磷为所有子集定义F⊂Ω一个样本空间Ω并假设你知道这个集合函数满足（2.7-2.9）。证明对于任意（不一定是不相交的）事件。
磷(F∪G)=磷(F)+磷(G)−磷(F∩G).
2. 描述子集的 sigma-fieldℜ由点或单例集生成。这个 sigma-field 是否包含形式的间隔(一种,b)为了b>一种 ?
3. 给定一个有限子集一种实线的ℜ，证明幂集一种和乙(一种)是相同的。重复一个可数无限的子集ℜ.
4. 鉴于离散样本空间Ω拥有n元素，表明幂集Ω由组成2n元素。
5. ∗让Ω=ℜ，实线，并考虑集合F的子集ℜ定义为所有形式的集合
⋃一世=0ķ(一种一世,b一世]∪⋃j=0米(Cj,dj]和
对于所有可能的非负整数选择ķ和米以及所有可能的实数选择一种一世<b一世,C一世<d一世. 如果ķ或者米为 0 ，则相应的联合被定义为空，因此空集本身具有给定的形式。换句话说，F包含这种形式的半开区间的所有可能的有限并集和这种半开区间的补集. 此表格的每一组都在F和每一组F有这种形式。证明F是一个子集的域Ω. 做F包含点？例如，是单例集0在F? 是F西格玛场？
6. 让Ω=[0,∞)是一个样本空间，让F是子集的 sigma 域Ω由所有形式的集合生成(n,n+1)为了n=1,2,…

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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|Computational Examples

This section is less detailed than its counterpart for discrete probability because generally engineers are more familiar with common integrals than with common sums. We confine the discussion to a few observations and to an example of a multidimensional probability computation.

The uniform pdf is trivially a valid pdf because it is nonnegative and its integral is simply the length of the the interval on which it is nonzero, $b-a$, divided by the length. For simplicity consider the case where $a=0$ and $b=1$ so that $b-a=1$. In this case the probability of any interval

within $[0,1)$ is simply the length of the interval. The mean is easily found to be
$$m=\int_{0}^{1} r d r=\left.\frac{r^{2}}{2}\right|{0} ^{1}=\frac{1}{2},$$ the second moment is $$m=\int{0}^{1} r^{2} d r=\left.\frac{r^{3}}{3}\right|{0} ^{1}=\frac{1}{3}$$ and the variance is $$\sigma^{2}=\frac{1}{3}-\left(\frac{1}{2}\right)^{2}=\frac{1}{12} .$$ The validation of the pdf and the mean, second moment, and variance of the exponential pdf can be found from integral tables or by the integral analog to the corresponding computations for the geometric pmf, as described in appendix B. In particular, it follows from (B.9) that $$\int{0}^{\infty} \lambda e^{-\lambda r} d r=1$$
from (B.10) that
$$m=\int_{0}^{\infty} r \lambda e^{-\lambda r} d r=\frac{1}{\lambda}$$
and
$$m^{(2)}=\int_{0}^{50} r^{2} \lambda e^{-\lambda r} d r=\frac{2}{\lambda^{2}}$$
and hence from (2.65)
$$\sigma^{2}=\frac{2}{\lambda^{2}}-\frac{1}{\lambda^{2}}=\frac{1}{\lambda^{2}} .$$
The moments can also be found by integration by parts.
The Laplacian pdf is simply a mixture of an exponential pdf and its reverse, so its properties follow from those of an exponential pdf. The details are left as an exercise.

The Gaussian pdf example is more involved. In appendix B, it is shown (in the development leading up to (B.15) that
$$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \sigma^{2}}} e^{-\frac{(x-m)^{2}}{2 \sigma^{2}}} d x=1 .$$
It is reasonably easy to find the mean by inspection. The function $g(x)=$ $(x-m) e^{-\frac{(x-m)^{2}}{2 \sigma^{2}}}$ is an odd function, i.e., it has the form $g(-x)=-g(x)$, and hence its integral is 0 if the integral exists at all.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Mass Functions as Densities

As in systems theory, discrete problems can be considered as continuous problems with the aid of the Dirac delta or unit impulse $\delta(t)$, a generalized function or singularity function (also, unfortunately, called a distribution) with the property that for any smooth function ${g(r) ; r \in \Re}$ and any $a \in \mathbb{R}$
$$\int g(r) \delta(r-a) d r=g(a)$$
Given a pmf $p$ defined on a subset of the real line $\Omega \subset \Re$, we can define a pdf $f$ by
$$f(r)=\sum p(\omega) \delta(r-\omega)$$
Thie ie indeed a pdf einee
\begin{aligned} \int f(r) d r &=\int\left(\sum p(\omega) \delta(r-\omega)\right) d r \ &=\sum p(\omega) \int \delta(r-\omega) d r \ &=\sum p(\omega)=1 . \end{aligned}

In a similar fashion, probabilies are computed as
\begin{aligned} \int 1_{F}(r) f(r) d r &=\int 1_{F}(r)\left(\sum p(\omega) \delta(r-\omega)\right) d r \ &=\sum p(\omega) \int 1_{F}(r) \delta(r-\omega) d r \ &=\sum p(\omega) 1_{F}(\omega)=P(F) . \end{aligned}
Given that discrete probability can be handled using the tools of continuous probability in this fashion, it is natural to inquire why not use pdf’s in both the discrete and continuous case. The main reason is simplicity, pmf’s and sums are usually simpler to handle and evaluate than pdf’s and integrals. Questions of existence and limits rarely arise, and the notation is simpler. In addition, the use of Dirac deltas assumes the theory of generalized functions in order to treat integrals involving Dirac deltas as if they were ordinary integrals, so additional mathematical machinery is required. As a result, this approach is rarely used in genuinely discrete problems. On the other hand, if one is dealing with a hybrid problem that has both discrete and continuous components, then this approach may make sense because it allows the use of a single probability function, a pdf, throughout.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Multidimensional pdf ’s

By considering multidimensional integrals we can also extend the construction of probabilities by integrals to finite-dimensional product spaces, e.g., bok $^{k}$.

Given the measurable space $\left(\Re^{k}, \mathcal{B}(\Re)^{k}\right)$, say we have a real-valued function $f$ on $R^{k}$ with the properties that
$$\begin{gathered} f(\mathbf{x}) \geq 0 ; \text { all } \mathbf{x}=\left(x_{0}, x_{1}, \ldots, x_{k-1}\right) \in x^{k} \ \int_{\mathfrak{R k}^{k}} f(\mathbf{x}) d \mathbf{x}=1 \end{gathered}$$
Then define a set function $P$ by
$$P(F)-\int_{F} f(\mathbf{x}) d \mathbf{x}, \text { all } F \in \mathcal{B}(\mathfrak{R})^{k},$$
where the vector integral is shorthand for the $k$ – dimensional integral, that is,
$$P(F)=\int_{\left(x_{0}, x_{1}, \ldots, x_{k-1}\right) \in F} f\left(x_{0}, x_{1}, \ldots, x_{k-1}\right) d x_{0} d x_{1} \ldots d x_{k-1}$$

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

σ2=2λ2−1λ2=1λ2.

∫−∞∞12σ2和−(X−米)22σ2dX=1.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Mass Functions as Densities

∫G(r)d(r−一种)dr=G(一种)

F(r)=∑p(ω)d(r−ω)
Thie ie 确实是一个 pdf einee
∫F(r)dr=∫(∑p(ω)d(r−ω))dr =∑p(ω)∫d(r−ω)dr =∑p(ω)=1.

∫1F(r)F(r)dr=∫1F(r)(∑p(ω)d(r−ω))dr =∑p(ω)∫1F(r)d(r−ω)dr =∑p(ω)1F(ω)=磷(F).

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Multidimensional pdf ’s

F(X)≥0; 全部 X=(X0,X1,…,Xķ−1)∈Xķ ∫RķķF(X)dX=1

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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|Continuous Probability Spaces

Continuous spaces are handled in a manner analogous to discrete spaces, but with some fundamental differences. The primary difference is that usually probabilities are computed by integrating a density function instead of eumming a maes function. The good news is that moet formulas look the same with integrals replacing sums. The bad news is that there are some underlying theoretical issues that require consideration. The problem is that integrals are themselves limits, and limits do not always exist in the sense of converging to a finite number. Because of this, some care will be needed to clarify when the resulting probabilities are well defined.
[2.14] Let $(\Omega, \mathcal{F})=(\Re, \mathcal{B}(\Re))$, the real line together with its Borel field. Suppose that we have a real-valued function $f$ on the real line that satisfies the following properties
$$\begin{gathered} f(r) \geq 0, \text { all } r \in \Omega \ \int_{\Omega} f(r) d r=1 \end{gathered}$$
that is, the function $f(r)$ has a well-defined integral over the real line. Define the set function $P$ by
$$P(F)=\int_{F} f(r) d r=\int 1_{F}(r) f(r) d r, F \in \mathcal{B}(\Re)$$
We note that a probability space defined as a probability measure on a Borel field is an example of a Borel space.

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

The first issue is fundamental: Does the integral of (2.56) make sense; i.e., is it well-defined for all events of interest? Suppose first that we take the common engineering approach and use Riemann integration – the form

of integration used in elementary calculus. Then the above integrals are defined at least for events $F$ that are intervals. This implies from the linearity properties of Riemann integration that the integrals are also welldefined for events $F$ that are finite unions of intervals. It is not difficult, however, to construct sets $F$ for which the indicator function $1_{F}$ is so nasty that the function $f(r) 1_{F}(r)$ does not have a Riemann integral. For example, suppose that $f(r)$ is 1 for $r \in[0,1]$ and 0 otherwise. Then the Riemann integral $\int 1_{F}(r) f(r) d r$ is not defined for the set $F$ of all irrational numbers, yet intuition should suggest that the set has probability 1 . This intuition reflects the fact that if all points are somehow equally probable, then since the unit interval contains an uncountable infinity of irrational numbers and only a countable infinity of rational numbers, then the probability of the former tet rhould bo one and that of the latter $\overline{0}$. This intuition in not reflected in the integral definition, which is not defined for either set by the Riemann approach. Thus the definition of (2.56) has a basic problem: The integral in the formula giving the probability measure of a set might not be well-defined.

A natural approach to escaping this dilemma would be to use the Riemann integral when possible, i.e., to define the probabilities of events that are finite unions of intervals, and then to obtain the probabilities of more complicated events by expressing them as a limit of finite unions of intervals, if the limit makes sense. This would hopefully give us a reasonable definition of a probability measure on a class of events much larger than the class of all finite unions of intervals. Intuitively, it should give us a probability measure of all sets that can be expressed as increasing or decreasing limits of finite unions of intervals.

This larger class is, in fact, the Borel field, but the Riemann integral has the unfortunate property that in general we cannot interchange limits and integration; that is, the limit of a sequence of integrals of converging functions may not be itself an integral of a limiting function.

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

The function $f$ used in (2.54) to $(2.56)$ is called a probability density function or $p d f$ since it is a nonnegative function that is integrated to find a total mass of probability, just as a mass density function in physics is integrated to find a total mass. Like a pmf, a pdf is defined only for points in $\Omega$ and not for sets. Unlike a pmf, a pdf is not in itself the probability of anything; for example, a pdf can take on values greater than one, while a pmf cannot. Under a pdf, points frequently have probability zero, even though the pdf is nonzero. We can, however, interpret a pdf as being proportional to a probability in the following sense. For a pmf we had
$$p(x)=P({x})$$
Suppose now that the sample space is the real line and that a pdf $f$ is defined. Let $F=[x, x+\Delta x)$, where $\Delta x$ is extremely small. Then if $f$ is sufficiently smooth, the mean value theorem of calculus implies that
$$P([x, x+\Delta x))=\int_{x}^{x+\Delta x} f(\alpha) d \alpha \approx f(x) \Delta x$$
Thus if a pdf $f(x)$ is multiplied by a differential $\Delta x$, it can be interpreted as (approximately) the probability of being within $\Delta x$ of $x$.

Both probability functions, the pmf and the pdf, can be used to define and compute a probability measure: The pmf is summed over all points in the event, and the pdf is integrated over all points in the event. If the sample space is the subset of the real line, both can be used to compute expectations such as moments.

Some of the most common pdf’s are listed below. As will be seen, these are indeed valid pdf’s, that is, they satisfy (2.54) and (2.55). The pdf’s are assumed to be 0 outside of the specified domain. $b, a, \lambda>0, m$, and $\sigma>0$ are parameters in $\Re$.
The uniform pdf. Given $b>a, f(r)=1 /(b-a)$ for $r \in[a, b]$.
The exponential pdf. $f(r)=\lambda e^{-\lambda r} ; r \geq 0$.
The doubly exponential (or Laplacian) pdf. $f(r)=\frac{\lambda}{2} e^{-\lambda|r|} ; r \in$ $\Re$.

The Gaussian (or Normal) pdf. $f(r)=\left(2 \pi \sigma^{2}\right)^{-1 / 2} \exp \left(\frac{-(r-m)^{2}}{2 \sigma^{2}}\right)$; $r \in{$. Since the density is completely described by two parameters: the mean $m$ and variance $\sigma^{2}>0$, it is common to denote it by $\mathcal{N}\left(m, \sigma^{2}\right)$.
Other univariate pdf’s may be found in Appendix C.
Just as we used a pdf to construct a probability measure on the space $(\Re, \mathcal{B}(\Re)$ ), we can also use it to define a probability measure on any smaller space $(A, B(A))$, where $A$ is a subset of $\Re$.

As a technical detail we note that to ensure that the integrals all behave as expected we must also require that $A$ itself be a Borel set of $\Re$ so that it is precluded from being too nasty a set. Such probability spaces can be considered to have a sample space of either $\Re$ or $A$, as convenient. In the former case events outside of $A$ will have zero probability.

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

[2.14] 让(Ω,F)=(ℜ,乙(ℜ)), 实线连同它的 Borel 场。假设我们有一个实值函数F在满足下列性质的实线上
F(r)≥0, 全部 r∈Ω ∫ΩF(r)dr=1

p(X)=磷(X)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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 Mass Functions

A function $p(\omega)$ satisfying $(2.30)$ and $(2.31)$ is called a probability mass function or $p m f$. It is important to observe that the probability mass function is defined only for points in the sample space, while a probability measure is defined for events, sets which belong to an event space. Intuitively, the probability of a set is given by the sum of the probabilities of the points as given by the pmf. Obviously it is much easier to describe the probability function than the probability measure since it need only be specified for points. The axioms of probability then guarantee that the probability function can be used to compute the probability measure. Note that given one, we can always determine the other. In particular, given the pmf $p$, we can construct $P$ using (2.32). Given $P$, we can find the corresponding pmf $p$ from the formula
$$p(\omega)=P({\omega}) .$$
We list below several of the most common examples of pmf’s. The reader should verify that they are all indeed valid pmf’s, that is, that they satisfy (2.30) and (2.31).

Thẻ binăry pmó. $\Omega={0,1} ; p(0)=1-p p, p(1)=p$, whẻrè $p$ is à parameter in $(0,1)$.

A uniform pmf. $\Omega=\mathcal{Z}{n}={0,1, \ldots, n-1}$ and $p(k)=1 / n ; k \in \mathcal{Z}{n}$.
The binomial pmf. $\Omega=\mathcal{Z}{n+1}={0,1, \ldots, n}$ and $$p(k)=\left(\begin{array}{c} n \ k \end{array}\right) p^{k}(1-p)^{n-k} ; k \in \mathcal{Z}{n+1}$$
where
$$\left(\begin{array}{l} n \ k \end{array}\right)=\frac{n !}{k !(n-k) !}$$
is the binomial coefficient.
The binary pmf is a probability model for coin flipping with a biased coin or for a single sample of a binary data stream. A uniform pmf on $Z_{6}$ can model the roll of a fair die. Observe that it would not be a good model for ASCII data since, for example, the letters $t$ and $e$ and the symbol for space have a higher probability than other letters. The binomial pmf is a probability model for the number of heads in $n$ successive independent flips of a biased coin, as will later be soen.

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

The various named pmf’s provide examples for computing probabilities and other expectations. Although much of this is prerequisite material, it does not hurt to collect several of the more useful tricks that arise in evaluating sums. The binary pmf is too simple to alone provide much interest, so first consider the uniform pmf on $\mathcal{Z}{n}$. This is trivially a valid pmf since it is nonnegative and sums to 1 . The probability of any set is simply $$P(F)=\frac{1}{n} \sum 1{F}(\omega)=\frac{#(F)}{n}$$

where $#(F)$ denotes the number of elements or points in the set $F$. The mean is given by
$$m=\frac{1}{n} \sum_{k=1}^{n} k=\frac{n+1}{2}$$
a standard formula easily verified by induction, as detailed in appendix B. The second moment is given by
\begin{aligned} m^{(2)} &=\frac{1}{n} \sum_{k=1}^{n} k^{2} \ &=\frac{(n+1)(2 n+1)}{6} \end{aligned}
as can also be verified by induction. The variance can be found by combining (2.43), (2.42), and (2.41).

The binomial pmf is more complicated. The first issue is to prove that it sums to one and hence is a valid pmf (it is obviously nonnegative). This is accomplished by recalling the binomial theorem from high school algebra:
$$(a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \ k \end{array}\right) a^{n} b^{n-k}$$
and setting $a=p$ and $b=1-p$ to write
\begin{aligned} \sum_{k=0}^{n} p(k) &=\sum_{k=0}^{n}\left(\begin{array}{c} n \ k \end{array}\right) p^{k}(1-p)^{n-k} \ &=(p+1-p)^{n} \ &=1 \end{aligned}
Finding moments is trickier here, and we shall later develop a much easier way to do this using exponential transforms. Nonetheless, it provides somẻ uiséful prácicicé tó compüté an exámplé sum, if only tó démonstraté later how much work can be avoided! Finding the mean requires evaluation of the sum
\begin{aligned} m &=\sum_{k=0}^{n} k\left(\begin{array}{c} n \ k \end{array}\right) p^{k}(1-p)^{n-k} \ &=\sum_{k=0}^{n} \frac{n !}{(n-k) !(k-1) !} p^{k}(1-p)^{n-k} \ &=\sum_{k=1}^{n} \frac{n !}{(n-k) !(k-1) !} p^{k}(1-p)^{n-k} \end{aligned}

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Multidimensional pmf ’s

While the foregoing ideas were developed for scalar sample spaces such as $\mathcal{Z}{+}$, they also apply to vector sample spaces. For example, if $A$ is a discrete space, then so is the vector space $A^{k}=\left{\right.$ all vectors $\mathrm{x}=\left(x{0}, \ldots x_{k-1}\right)$ with $\left.x_{i} \in A, i=0,1, \ldots, k-1\right}$. A common example of a pmf on vectors is the product pmf of the following example.
[2.15] The product pmf.
Let $p_{i} ; i=0,1, \ldots, k-1$, be a collection of one-dimensional pmf’s; that is, for each $i=0,1, \ldots, k-1 p_{i}(k) ; r \in A$ satisfies (2.30) and (2.31). Define the product $k=$ dimensional pmf $p$ on $A^{k}$ by
$$p(\mathbf{x})=p\left(x_{0}, x_{1}, \ldots, x_{k-1}\right)=\prod_{i=0}^{k-1} p_{i}\left(x_{i}\right)$$

As a more specific example, suppose that all of the marginal pmf’s are the same and are given by a Bernoulli pmf:
$$p(x)=p^{x}(1-p)^{1-x} ; x=0,1 .$$
Then the corresponding product pmf for a $k$ dimensional vector becomes
\begin{aligned} p\left(x_{0}, x_{1}, \ldots, x_{k-1}\right) &=\prod_{i=0}^{k-1} p^{x_{i}}(1-p)^{1-x_{i}} \ &=p^{w\left(x_{0}, x_{1}, \ldots, x_{k-1}\right)}(1-p)^{k-w\left(x_{0}, x_{1}, \ldots, x_{k-1}\right)} \end{aligned}
where $w\left(x_{0}, x_{1}, \ldots, x_{k-1}\right)$ is the number of ones occurring in the binary $k$-tuple $x_{0}, x_{1}, \ldots, x_{k-1}$, the Hamming weight of the vector.

p(ω)=磷(ω).

(n ķ)=n!ķ!(n−ķ)!

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

(一种+b)n=∑ķ=0n(n ķ)一种nbn−ķ

∑ķ=0np(ķ)=∑ķ=0n(n ķ)pķ(1−p)n−ķ =(p+1−p)n =1

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Multidimensional pmf ’s

[2.15] 产品 pmf。

p(X)=p(X0,X1,…,Xķ−1)=∏一世=0ķ−1p一世(X一世)

p(X)=pX(1−p)1−X;X=0,1.

p(X0,X1,…,Xķ−1)=∏一世=0ķ−1pX一世(1−p)1−X一世 =p在(X0,X1,…,Xķ−1)(1−p)ķ−在(X0,X1,…,Xķ−1)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|A Single Coin Flip

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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|A Single Coin Flip

The original example of a spinning wheel is continuous in that the sample space consists of a continum of possible outcomes, all points in the unit interval. Sample spaces can also be discrete, as is the case of modeling a single flip of a “fair” coin with heads labeled ” 1 ” and tails labeled ” 0 “, i.e., heads and tails are equally likely. The sample space in this example is $\Omega={0,1}$ and the probability for any event or subset of $\omega$ can be defined in a reasonable way by
$$P(F)=\sum_{r \in F} p(r)$$

or, equivalently,
$$P(F)=\sum 1_{F}(r) p(r),$$
where now $p(r)=1 / 2$ for each $r \in \Omega$. The function $p$ is called a probability mass function or $p m f$ because it is summed over points to find total probability, just as point masses are summed to find total mass in physics. Be cautioned that $P$ is defined for sets and $p$ is defined only for points in the sample space. This can be confusing when dealing with one-point or singleton sets, for example
\begin{aligned} &P({0})=p(0) \ &P({1})=p(1) \end{aligned}
This may seem too much work for such a little example, but keep in mind that the goal is a formulation that will work for far more complicated and interesting examples. This example is different from the spinning wheel in that the sample space is discrete instead of continuous and that the probabilities of events are defined by sums instead of integrals, as one should expect when doing discrete math. It is easy to verify, however, that the basic properties (2.7)-(2.9) hold in this case as well (since sums behave like integrals), which in turn implies that the simple properties (a)-(d) also hold.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|A Single Coin Flip as Signal Processing

The coin flip example can also be derived in a very different way that provides our first example of signal processing. Consider again the spinning pointer so that the sample space is $\Omega$ and the probability measure $P$ is described by (2.2) using a uniform pdf as in (2.4). Performing the experiment by spinning the pointer will yield some real number $r \in[0,1)$. Define a measurement $q$ made on this outcome by
$$q(r)= \begin{cases}1 & \text { if } r \in[0,0.5] \ 0 & \text { if } r \in(0.5,1)\end{cases}$$
This function can also be defined somewhat more economically as
$$q(r)=1_{[0,0.5]}(r)$$
This is an example of a quantizer, an operation that maps a continuots value into a discrete one. Quantization is an example of signal processing since it is a function or mapping defined on an input space, here $\Omega=[0,1)$

or $\Omega=\Re$, producing a value in some output space, here a binary space $\Omega_{g}={0,1}$. The dependence of a function on its input space or domain of definztion $\Omega 2$ and its output space or range $\Omega_{g}$, is often denoted by $q$ : $\Omega \rightarrow \Omega_{g}$. Although introduced as an example of simple signal processing, the usual name for a real-valued function defined on the sample space of a probability space is a random varzable. We shall see in the next chapter that there is an extra technical condition on functions to merit this name. but that is a detail that can be postponed.

The output space $\Omega_{g}$ can be considered as a new sample space, the space corresponding to the possible values seen by an observer of the output of the quantizer (an observer who might not have access to the original space). If we know both the probability measure on the input space and the function, then in theory we should be able to describe the probability measure that the output space inherits from the input space. Since the output space is discrete, it should be described by a pmf, say $p_{q}$. Since there are only two points, we need only find the value of $p_{q}(1)$ (or $p_{q}(0)$ since $\left.p_{q}(0)+p_{q}(1)=1\right)$. An output of 1 is seen if and only if the input sample point lies in $[0,0.5]$, so it follows easily that $p_{q}(1)=P([0,0.5])=\int_{0}^{0.5} f(r)$, dr $=0.5$, exactly the value assumed for the fair coin flip model. The pmf $p_{q}$ implies a probability measure on the output space $\Omega_{g}$ by
$$P_{q}(F)=\sum_{\omega \in F} p_{q}(\omega),$$

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Abstract vs. Concrete

It may seem strange that the axioms of probability deal with apparently abstract ideas of measures instead of corresponding physical intuition that the probability tells you something about the fraction of times specific events will occur in a sequence of trials, such as the relative frequency of a pair of dice summing to seven in a sequence of many roles, or a decision algorithm correctly detecting a single binary symbol in the presence of noise in a transmitted data file. Such real world behavior can be quantified by the idea of a relative frequency, that is, suppose the output of the $n$th trial of a sequence of trials is $x_{n}$ and we wish to know the relative frequency that $x_{n}$ takes on a particular value, say $a$. Then given an infinite sequence of trials $x=\left{x_{0}, x_{1}, x_{2}, \ldots\right}$ we could define the relative frequency of $a$ in $x$ by
$$r_{a}(x)=\lim {n \rightarrow \infty} \frac{\text { number of } k \in{0,1, \ldots, n-1} \text { for which } x{k}=a}{n}$$
For example, the relative frequency of heads in an infinite sequence of fair coin flips should be $0.5$, the relative frequency of rolling a pair of fair dice and having the sum be 7 in an infinite sequence of rolls should be $1 / 6$ since the pairs $(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)$ are equally likely and form 6 of the possible 36 pairs of outcomes. Thus one might suspect that to make a rigorous theory of probability requires only a rigorous definition of probabilities as such limits and a reaping of the resulting benefits. In fact much of the history of theoretical probability consisted of attempts to acoomplish this, but unfortunately it does not work. Such limits might not exist, or they might exist and not converge to the same thing for different repetitions of the same experiment. Even when the limits do exist there is no guarantee they will behave as intuition would suggest when one tries to do calculus with probabilities, to compute probabilities of complicated events from those of simple related events. Attempts to get around these problems uniformly failed and probability was not put on a rigorous basis until the axiomatic approach was completed by Kolmogorov. The axioms do, however, capture certain intuitive aspects of relative frequencies. Relative frequencies are nonnegative, the relative frequency of the entire set of possible outcomes is one, and relative frequencies are additive in the sense that the relative frequency of the symbol $a$ or the symbol $b$ occurring. $r_{a \cup b}(x)$, is clearly $r_{a}(x)+r_{b}(x)$. Kolmogorov realized that beginning with simple axioms could lead to rigorous limiting results of the type needed, while there was no way to begin with the limiting results as part of the axioms. In fact it is the fourth axiom, a limiting version of additivity, that plays the key role in making the asymptotics work.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|A Single Coin Flip as Signal Processing

q(r)={1 如果 r∈[0,0.5] 0 如果 r∈(0.5,1)

q(r)=1[0,0.5](r)

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Abstract vs. Concrete

r一种(X)=林n→∞ 数量 ķ∈0,1,…,n−1 为此 Xķ=一种n

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|Organization of the Book

statistics-lab™ 为您的留学生涯保驾护航 在代写随机信号处理Statistical Signal Processing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。我们的专家在代写随机信号处理Statistical Signal Processing代写方面经验极为丰富，各种代写随机信号处理Statistical Signal Processing相关的作业也就用不着说。

• 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|Organization of the Book

Chapter 2 provides a careful development of the fundamental concept of probability theory – a probability space or experiment. The notions of sample space, event space, and probability meastre are introduced, and several examples are toured. Independence and elementary conditional probability are developed in some detail. The ideas of signal processing and of random variables are introduced briefly as functions or operations on the output of an experiment. This in turn allows mention of the idea of expectation at an early stage as a generalization of the description of probabilities by sums or integrals.

Chapter 3 treats the theory of measurements made on experiments: random variables, which are scalar-valued measurements; random vectors, which are a vector or finite collection of measurements; and random processes, which can be viewed as sequences or waveforms of measurements. Random variables, vectors, and processes can all be viewed as forms of signal processing: each operates on “inputs,” which are the sample points of a probability space, and produces an “output,” which is the resulting sample value of the random variable, vector, or process. These output points together constitute an output sample space, which inherits its own probability measure from the structure of the measurement and the underlying experiment. As a result, many of the basic properties of random variables, vectors, and processes follow from those of probability spaces. Probability distributions are introduced along with probability mass functions, probability density functions, and cumulative distribution functions. The basic derived distribution method is described and demonstrated by example. A wide variety of examples of random variables, vectors, and processes are treated.

Chapter 4 develops in depth the ideas of expectation, averages of random objects with respect to probability distributions. Also called proba-bilistic averages, statistical averages, and ensemble averages, expectations can be thought of as providing simple but important parameters describing probability distributions. A variety of specific averages are considered, including mean, variance, characteristic functions, correlation, and covariance. Several examples of unconditional and conditional expectations and their properties and applications are provided. Perhaps the most important application is to the statement and proof of laws of large numbers or ergodic theorems, which relate long term sample average behavior of random processes to expectations. In this chapter laws of large numbers are proved for simple, but important, classes of random processes. Other important applications of expectation arise in performing and analyzing signal processing applications such as detecting, classifying, and estimating data. Minimum mean squared nonlinear and linear estimation of scalars and vectors is treated in some detail, showing the fundamental connections among conditional expectation, optimal estimation, and second order moments of random variables and vectors.

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

The theory of random processes is a branch of probability theory and probability theory is a special case of the branch of mathematics known as measure theory. Probability theory and measure theory both concentrate on functions that assign real numbers to certain sets in an abstract space according to certain rules. These set functions can be viewed as measures of the size or weight of the sets. For example, the precise notion of area in two-dimensional Euclidean space and volume in three-dimensional space are both examples of measures on sets. Other measures on sets in three dimensions are mass and weight. Observe that from elementary calculus we can find volume by integrating a constant over the set. From physics we can find mass by integrating a mass density or summing point masses over a set. In both cases the set is a region of three-dimensional space. In a similar manner, probabilities will be computed by integrals of densities of probability or sums of “point masses” of probability.

Both probability theory and measure theory consider only nonnegative real-valued set functions. The value assigned by the function to a set is called the probability or the measure of the set, respectively. The basic difference between probability theory and measure theory is that the former considers only set functions that are normalized in the sense of assigning the value of 1 to the entire abstract space, corresponding to the intuition that the abstract space contains every possible outcome of an experiment and hence should happen with certainty or probability 1. Subsets of the space have some uncertainty and hence have probability less than $1 .$

Probability theory begins with the concept of a probability space, which is a collection of three items.

## 统计代写|随机信号处理作业代写Statistical Signal Processing代考|A Uniform Spinning Pointer

Suppose that Nature (or perhaps Tyche, the Greek Goddess of chance) spins a pointer in a circle as depicted in Figure 2.1. When the pointer stops it can point to any number in the unit interval $[0,1) \triangleq{r: 0 \leq r<1}$. We call $[0,1)$ the sample space of our experiment and denote it by a capital Greek omega, $\Omega$. What can we say about the probabilities or chances of particular events or outcomes occurring as a result of this experiment? The sorts of events of interest are things like “the pointer points to a number between 0 and .5” (which one would expect should have probability $0.5$ if the wheel is indeed fair) or “the pointer does not lie between $0.75$ and $1^{“}$ (which should have a probability of $0.75$ ). Two assumptions are implicit here. The first is that an “outcome” of the experiment or an “event” to which we can assign a probability is simply a subset of $[0,1)$. The second assumption is that the probability of the pointer landing in any particular interval of the sample space is proportional to the length of the interval. This should seem reasonable if we indeed believe the spinning pointer to be “fair” in the sense of not favoring any outcomes over any others. The bigger a region of the circle, the more likely the pointer is to end up in that region. We can formalize this by stating that for any interval $[a, b]={r: a \leq r \leq b}$ with $0 \leq a \leq b<1$ we have that the probability of the event “the pointer lands

in the interval $[a, b]^{\top}$ is
$$P([a, b])=b-a$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。