## 数学代写|概率论代写Probability theory代考|MATHS2103

2023年1月3日

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## 数学代写|概率论代写Probability theory代考|Continuous random variables

Asmentioned, in addition to discrete random variables investigated in Section 4.3, Ithere is another type of random variables with innumerable or continuous possible values. These are called as the continuous random variables. Some examples of this type are as follows:
The lifetime of light bulbs produced in a factory
The weight of each cement bag produced in a related factory
The jump magnitude of an athlete in a triple jump competition
The daily amount of rain in a certain district
One of the properties of this type of random variables is that they usually possess a unit of measurement. In other words, unlike some discrete random variables that have no units of measurement (like the random variables of Example 1.2), continuous random variables usually have a unit of measurement such as the minute, gram, meter, square meter, and to name but a few.

Discrete random variables are defined by the probability function or $P(X=x)$. On the contrary, continuous random variables are not defined by $P(X=x)$. This is because of the fact that the probability function of these random variables converges to zero in each point. For instance, it can be shown the probability that the lifetime of a light bulb takes on a value in the range of $10000 \pm 1$ (measured in seconds) is negligible. Now, since this interval contains innumerable number of points, it can be concluded the probability that the lifetime of the light bulb is exactly equal to 10000 converges to zero. Therefore, instead of the probability function for continuous random variables, we define the probability density function or briefly density function. In this type of random variables, the probability density function is defined as follows.

## 数学代写|概率论代写Probability theory代考|Mixed random variables

Asmentioned before, except for the discrete and continuous random variables, Athere is another type of random variables consisting of a combination of discrete and continuous random variables called mixed random variables. In this type, some parts of the range are discrete and some others are continuous. For the part with discrete range, the probability mass function is defined, and for the part with continuous range, the probability density function is defined. For instance, note the following examples:

Suppose that waiting time (measured in seconds) of cars at a stoplight follows a mixed random variable. Cars meet green light with the probability of $\frac{1}{2}$ and their waiting time at the spotlight equals zero. Otherwise, their waiting time follows density function $f(x)=\frac{1}{60} ; 0<x<30$. In such a case, what percentage of cars wait less than 15 seconds at the spotlight?

Solution. In this mixed random variable, the probability function for discrete values and the density function for continuous values are as follows:
$$\left{\begin{array}{lc} P(x)=\frac{1}{2} ; \quad x=0 \ f(x)=\frac{1}{60} ; \quad 0<x<30 \end{array}\right.$$
Therefore, to calculate the probability that one car waits less than 15 seconds, we have:
$$P(X<15)=P(X=0)+P(0<X<15)=\frac{1}{2}+\int_0^{15} \frac{1}{60} d x=\frac{1}{2}+\frac{1}{4}=0.75$$
Hence, 75 percent of cars wait less than 15 seconds at the spotlight.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Mixed random variables

Veft {
$$P(x)=\frac{1}{2} ; \quad x=0 f(x)=\frac{1}{60} ; \quad 0<x<30$$

Therefore, tocalculatetheprobabilitythatone
\begin{aligned} & P(X<15)=P(X=0)+P(0<X<15)=\backslash \text { frac }{1}{2}+\backslash \text { int_ } 0^{\wedge}{15} \ & \text { Ifrac }{1} 60} d x=\backslash \text { frac }{1} 2}+\backslash \operatorname{frac}{1}{4}=0.75 \end{aligned}
$\$ \

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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。