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# 金融代写|风险理论投资组合代写MARKET RISK, MEASURES AND PORTFOLIO代考|CONTINUOUS PROBABILITY DISTRIBUTIONS

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## 金融代写|风险理论投资组合代写MARKET RISK, MEASURES AND PORTFOLIO代考|Probability Distribution Function, Probability Density Function, and Cumulative Distribution Function

A probability distribution function P assigns a probability $P(A)$ for every event A, that is, of realizing a value for the random value in any specified subset A of the sample space. For example, a probability distribution function can assign a probability of realizing a monthly return that is negative or the probability of realizing a monthly return that is greater than $0.5 \%$ or the probability of realizing a monthly return that is between $0.4 \%$ and $1.0 \%$.

To compute the probability, a mathematical function is needed to represent the probability distribution function. There are several possibilities of representing a probability distribution by means of a mathematical function. In the case of a continuous probability distribution, the most popular way is to provide the so-called probability density function or simply density function.

In general, we denote the density function for the random variable X as $f_{X}(x)$. Note that the letter $x$ is used for the function argument and the index denotes that the density function corresponds to the random variable $X$. The letter $x$ is the convention adopted to denote a particular value for the random variable. The density function of a probability distribution is always nonnegative and as its name indicates: Large values for $f_{X}(x)$ of the density function at some point x imply a relatively high probability of realizing a value in the neighborhood of x, whereas $f_{X}(x)=0$ for all $x$ in some interval a, bimplies that the probability for observing a realization in a, b is zero.

## 金融代写|风险理论投资组合代写MARKET RISK, MEASURES AND PORTFOLIO代考|The Normal Distribution

The class of normal distributions, or Gaussian distributions, is certainly one of the most important probability distributions in statistics and due to some of its appealing properties also the class which is used in most applications in finance. Here we introduce some of its basic properties.

The random variable $X$ is said to be normally distributed with parameters $\mu$ and $\sigma$, abbreviated by $X \in N\left(\mu, \sigma^{2}\right)$, if the density of the random variable is given by the formula,
$$f_{X}(x)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}}, x \in \mathbb{R} .$$

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio代考|Exponential Distribution

The exponential distribution is popular, for example, in queuing theory when we want to model the time we have to wait until a certain event takes place. Examples include the time until the next client enters the store, the time until a certain company defaults or the time until some machine has a defect.

As it is used to model waiting times, the exponential distribution is concentrated on the positive real numbers and the density function $f$ and the cumulative distribution function $F$ of an exponentially distributed random variable $\tau$ possess the following form:
$$f_{\tau}(x)=\frac{1}{\beta} e^{-\frac{x}{\beta}}, x>0$$
and
$$F_{\tau}(x)=1-e^{-\frac{x}{\beta}}, x>0 .$$

## 金融代写|风险理论投资组合代写MARKET RISK, MEASURES AND PORTFOLIO代考|THE NORMAL DISTRIBUTION

FX(X)=12圆周率σ2和−(X−μ)22σ2,X∈R.

## 金融代写|风险理论投资组合代写MARKET RISK, MEASURES AND PORTFOLIO代考|EXPONENTIAL DISTRIBUTION

Fτ(X)=1b和−Xb,X>0

Fτ(X)=1−和−Xb,X>0.

## 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。