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

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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|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

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## MATLAB代写

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