# 统计代写|随机过程代写stochastic process代考|RANDOM SELECTION OF POINTS FROM INTERVALS

#### Doug I. Jones

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## 统计代写|随机过程代写stochastic process代考|RANDOM SELECTION OF POINTS FROM INTERVALS

In Section 1.6, we showed that the probability of the occurrence of any particular point in a random selection of points from an interval $(a, b)$ is 0 . This implies immediately that if $[\alpha, \beta] \subseteq(a, b)$, then the events that the point falls in $[\alpha, \beta],(\alpha, \beta),[\alpha, \beta)$, and $(\alpha, \beta]$ are all equiprobable. Now consider the intervals $\left(a, \frac{a+b}{2}\right)$ and $\left(\frac{a+b}{2}, b\right)$; since $\frac{a+b}{2}$ is the midpoint of $(a, b)$, it is reasonable to assume that
$$p_1=p_2,$$
where $p_1$ is the probability that the point belongs to $\left(a, \frac{a+b}{2}\right)$ and $p_2$ is the probability that it belongs to $\left(\frac{a+b}{2}, b\right)$. The events that the random point belongs to $\left(a, \frac{a+b}{2}\right)$ and $\left(\frac{a+b}{2}, b\right)$ are mutually exclusive and
$$\left(a, \frac{a+b}{2}\right) \cup\left[\frac{a+b}{2}, b\right)=(a, b) ;$$
therefore,
$$p_1+p_2=1$$
This relation and (1.5) imply that
$$p_1=p_2=1 / 2$$

## 统计代写|随机过程代写stochastic process代考|WHAT IS SIMULATION?

Solving a scientific or an industrial problem usually involves mathematical analysis and/or simulation. To perform a simulation, we repeat an experiment a large number of times to assess the probability of an event or condition occurring. For example, to estimate the probability of at least one 6 occurring within four rolls of a die, we may do a large number of experiments rolling a die four times and calculate the number of times that at least one 6 is obtained. Similarly, to estimate the fraction of time that, in a certain bank all the tellers are busy, we may measure the lengths of such time intervals over a long period $X$, add them, and then divide by $X$. Clearly, in simulations, the key to reliable answers is to perform the experiment a large number of times or over a long period of time, whichever is applicable. Since manually this is almost impossible, simulations are carried out by computers. Only computers can handle millions of operations in short periods of time.

To simulate a problem that involves random phenomena, generating random numbers from the interval $(0,1)$ is essential. In almost every simulation of a probabilistic model,we will need to select random points from the interval $(0,1)$. For example, to simulate the experiment of tossing a fair coin, we draw a random number from $(0,1)$. If it is in $(0,1 / 2)$, we say that the outcome is heads, and if it is in $[1 / 2,1)$, we say that it is tails. Similarly, in the simulation of die tossing, the outcomes $1,2,3,4,5$, and 6 , respectively, correspond to the events that the random point from $(0,1)$ is in $(0,1 / 6),[1 / 6,1 / 3),[1 / 3,1 / 2),[1 / 2,2 / 3),[2 / 3,5 / 6)$, and $[5 / 6,1)$.

As discussed in Section 1.7, choosing a random number from a given interval is, in practice, impossible. In real-world problems, to perform simulation we use pseudorandom numbers instead. To generate $n$ pseudorandom numbers from a uniform distribution on an interval $(a, b)$, we take an initial value $x_0 \in(a, b)$, called the seed, and construct a function $\psi$ so that the sequence $\left{x_1, x_2, \ldots, x_n\right} \subset(a, b)$ obtained recursively from
$$x_{i+1}=\psi\left(x_i\right), \quad 0 \leq i \leq n-1,$$
satisfies certain statistical tests for randomness. (Choosing the tests and constructing the function $\psi$ are complicated matters beyond the scope of this book.) The function $\psi$ takes a seed and generates a sequence of pseudorandom numbers in the interval $(a, b)$. Clearly, in any pseudorandom number generating process, the numbers generated are rounded to a certain number of decimal places. Therefore, $\psi$ can only generate a finite number of pseudorandom numbers, which implies that, eventually, some $x_j$ will be generated a second time. From that point on, by (1.7), a pitfall is that the same sequence of numbers that appeared after $x_j$ ‘s first appearance will reappear. Beyond that point, numbers are not effectively random. One important aspect of the construction of $\psi$ is that the second appearance of any of the $x_j$ ‘s is postponed as long as possible.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|RANDOM SELECTION OF POINTS FROM INTERVALS

$$p_1=p_2,$$

$$\left(a, \frac{a+b}{2}\right) \cup\left[\frac{a+b}{2}, b\right)=(a, b) ;$$

$$p_1+p_2=1$$

$$p_1=p_2=1 / 2$$

## 统计代写|随机过程代写stochastic process代考|WHAT IS SIMULATION?

$$x_{i+1}=\psi\left(x_i\right), \quad 0 \leq i \leq n-1,$$

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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。

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