## 数学代写|概率论代写Probability theory代考|STAT7614

2023年1月3日

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## 数学代写|概率论代写Probability theory代考|Cumulative distribution function

Asmentioned before, for discrete random variables and continuous random Avariables, the probability function and probability density function are respectively defined. Moreover, for mixed random variables, the probability function is defined for discrete parts and the density function is defined for continuous parts. However, there is a function known as the cumulative distribution function commonly defined for all three types of random variables, which is defined as follows:
$$F_X(a)=P(X \leq a)$$
That is, the cumulative distribution function of the random variable $X$ at point ” $a$ ” is equal to the probability that the random variable $X$ adopts a value less than or equal to ” $a$ “.

For example, suppose that the random variable $X$ adopts values 1,2 , and 3 with respective probabilities $\frac{1}{4}, \frac{1}{2}$, and $\frac{1}{4}$. In such a case, the probability function of this random variable is as follows: To obtain the cumulative distribution function of this variable, according to the definition of this function, we should obtain for each point like $x$ the probability that $X$ takes on values less than or equal to it. Namely, it is evident that the function for $x=2$ is equal to $F_X(2)=P(X \leq 2)=\frac{3}{4}$, for $x=1.5$ is equal to $F_X(1.5)=P(X \leq 1.5)=$ $\frac{1}{4}$, and for $x=0$ is equal to $F_X(0)=P(X \leq 0)=0$. Likewise, the cumulative distribution function value can be obtained for all the points and its function is as follows:

The cumulative distribution function of the continuous random variable $X$ is also obtained as follows:
$$F_X(a)=P(X \leq a)=P(X<a)=\int_{-\infty}^a f_X(x) d x$$
The above equation is simply the area under the density function curve in an interval less than or equal to $a$.

## 数学代写|概率论代写Probability theory代考|Some important values of random variables

There are some values in random variables that are important in practice. Some of these values are median, mode, and mean. However, it can be shown that all of these values necessarily do not exist. We introduce the median and mode in this chapter followed by the explanation of the mean of a random variable in the next chapter.

Definition: The median of a random variable is a value that its less-than-or-equal-to probability is equal to at least $0.5$, and that its greater-than-or-equal-to probability is equal to at least $0.5$.
$$P(X \geq m) \geq 0.5, \quad P(X \leq m) \geq 0.5$$
Nevertheless, it can simply be shown that, for continuous random variables, the median is a value that its less-than-or-equal-to probability is equal to $0.5$, and that its greater-than-or-equal-to probability is equal to $0.5$.

Definition: The mode of a random variable is a value with the highest probability. For continuous random variables, the mode is a value with the highest probability density.

Solution. It is evident that the mode of the random variable $X$ is equal to 2 , but random variable $Y$ has two modes with the values of 2 and 3 .

Considering the definition of median for the random variable $X$, the only value that its less-than-or-equal-to probability and that its greater-than-or-equal-to probability are equal to at least $0.5$ is value 2 . This means that the other values do not have this property.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Cumulative distribution function

$$F_X(a)=P(X \leq a)$$

$$F_X(a)=P(X \leq a)=P(X<a)=\int_{-\infty}^a f_X(x) d x$$

## 数学代写|概率论代写Probability theory代考|Some important values of random variables

$$P(X \geq m) \geq 0.5, \quad P(X \leq m) \geq 0.5$$

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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。