Posted on Categories:Stochastic Porcesses, 数学代写, 随机过程

数学代写|随机过程Stochastic Porcess代考|INDEPENDENCE

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数学代写|随机过程代写Stochastic Porcess代考|INDEPENDENCE

Let $A$ and $B$ be two events of a sample space $S$, and assume that $P(A)>0$ and $P(B)>0$. We have seen that, in general, the conditional probability of $A$ given $B$ is not equal to the probability of $A$. However, if it is, that is, if $P(A \mid B)=P(A)$, we say that $A$ is independent of $B$. This means that if $A$ is independent of $B$, knowledge regarding the occurrence of $B$ does not change the chance of the occurrence of $A$. The relation $P(A \mid B)=P(A)$ is equivalent to the relations $P(A B) / P(B)=P(A), P(A B)=P(A) P(B), P(B A) / P(A)=P(B)$, and $P(B \mid A)=P(B)$. The equivalence of the first and last of these relations implies that if $A$ is independent of $B$, then $B$ is independent of $A$. In other words, if knowledge regarding the occurrence of $B$ does not change the chance of occurrence of $A$, then knowledge regarding the occurrence of $A$ does not change the chance of occurrence of $B$. Hence independence is a symmetric relation on the set of all events of a sample space. As a result of this property, instead of making the definitions ” $A$ is independent of $B$ ” and ” $B$ is independent of $A$,” we simply define the concept of the “independence of $A$ and $B$.” To do so, we take $P(A B)=$ $P(A) P(B)$ as the definition. We do this because a symmetrical definition relating $A$ and $B$ does not readily follow from either of the other relations given [i.e., $P(A \mid B)=P(A)$ or $P(B \mid A)=P(B)]$. Moreover, these relations require either that $P(B)>0$ or $P(A)>0$, whereas our definition does not.
Definition 3.3 Two events $A$ and $B$ are called independent if
$$P(A B)=P(A) P(B) .$$
If two events are not independent, they are called dependent. If $A$ and $B$ are independent, we say that ${A, B}$ is an independent set of events.

Note that in this definition we did not require $P(A)$ or $P(B)$ to be strictly positive. Hence by this definition any event $A$ with $P(A)=0$ or 1 is independent of every event $B$ (see Exercise 16).

数学代写|随机过程代写Stochastic Porcess代考|RANDOM VARIABLES

In real-world problems we are often faced with one or more quantities that do not have fixed values. The values of such quantities depend on random actions, and they usually change from one experiment to another. For example, the number of babies born in a certain hospital each day is not a fixed quantity. It is a complicated function of many random factors that vary from one day to another. So are the following quantities: the arrival time of a bus at a station, the sum of the outcomes of two dice when thrown, the amount of rainfall in Seattle during a given year, the number of earthquakes that occur in California per month, and the weight of grains of wheat grown on a certain plot of land (it varies from one grain to another). In probability, quantities introduced in these diverse examples are called random variables. The numerical values of random variables are unknown. They depend on random elements occurring at the time of the experiment and over which we have no control. For example, if in rolling two fair dice, $X$ is the sum, then $X$ can only assume the values $2,3,4, \ldots, 12$ with the following probabilities:
\begin{aligned} & P(X=2)=P({(1,1)})=1 / 36, \ & P(X=3)=P({(1,2),(2,1)})=2 / 36, \ & P(X=4)=P({(1,3),(2,2),(3,1)})=3 / 36, \end{aligned}
and, similarly,
\begin{tabular}{c|cccccccc}
Sum, $i$ & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \
\hline$P(X=i)$ & $4 / 36$ & $5 / 36$ & $6 / 36$ & $5 / 36$ & $4 / 36$ & $3 / 36$ & $2 / 36$ & $1 / 36$
\end{tabular}
Clearly, ${2,3,4, \ldots, 12}$ is the set of possible values of $X$. Since $X \in{2,3,4, \ldots, 12}$, we should have $\sum_{i=2}^{12} P(X=i)=1$, which is readily verified. The numerical value of a random variable depends on the outcome of the experiment. In this example, for instance, if the outcome is $(2,3)$, then $X$ is 5 , and if it is $(5,6)$, then $X$ is $11 . X$ is not defined for points that do not belong to $S$, the sample space of the experiment. Thus $X$ is a real-valued function on $S$. However, not all real-valued functions on $S$ are considered to be random variables. For theoretical reasons, it is necessary that the inverse image of an interval in $\mathbf{R}$ be an event of $S$, which motivates the following definition.

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。