统计代写|金融统计代写Mathematics with Statistics for Finance代考|Basic Statistics

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|金融统计代写Mathematics with Statistics for Finance代考|Population and Sample Data

If you wanted to know the mean age of people working in your firm, you would simply ask every person in the firm his or her age, add the ages together, and divide by the number of people in the firm. Assuming there are $n$ employees and $a_{i}$ is the age of the $i$ th employee, then the mean, $\mu$, is simply:
$$\mu=\frac{1}{n} \sum_{i=1}^{n} a_{i}=\frac{1}{n}\left(a_{1}+a_{2}+\cdots+a_{n-1}+a_{n}\right)$$
It is important at this stage to differentiate between population statistics and sample statistics. In this example, $\mu$ is the population mean. Assuming nobody lied about his or her age, and forgetting about rounding errors and

other trivial details, we know the mean age of people in your firm exactly. We have a complete data set of everybody in your firm; we’ve surveyed the entire population.

This state of absolute certainty is, unfortunately, quite rare in finance. More often, we are faced with a situation such as this: estimate the mean return of stock $\mathrm{ABC}$, given the most recent year of daily returns. In a situation like this, we assume there is some underlying data generating process, whose statistical properties are constant over time. The underlying process still has a true mean, but we cannot observe it directly. We can only estimate that mean based on our limited data sample. In our example, assuming $n$ returns, we estimate the mean using the same formula as before:
$$\hat{\mu}=\frac{1}{n} \sum_{i=1}^{n} r_{i}=\frac{1}{n}\left(r_{1}+r_{2}+\cdots+r_{n-1}+r_{n}\right)$$
where $\hat{\mu}$ (pronounced “mu hat”) is our estimate of the true mean based on our sample of $n$ returns. We call this the sample mean.

The median and mode are also types of averages. They are used less frequently in finance, but both can be useful. The median represents the center of a group of data; within the group, half the data points will be less than the median, and half will be greater. The mode is the value that occurs most frequently.

统计代写|金融统计代写Mathematics with Statistics for Finance代考|Discrete Random Variables

For a discrete random variable, we can also calculate the mean, median, and mode. For a random variable, $X$, with possible values, $x_{i}$, and corresponding probabilities, $p_{i}$, we define the mean, $\mu$, as:
$$\mu=\sum_{i=1}^{n} p_{i} x_{i}$$

The equation for the mean of a discrete random variable is a special case of the weighted mean, where the outcomes are weighted by their probabilities, and the sum of the weights is equal to one.

The median of a discrete random variable is the value such that the probability that a value is less than or equal to the median is equal to $50 \%$. Working from the other end of the distribution, we can also define the median such that $50 \%$ of the values are greater than or equal to the median. For a random variable, $X$, if we denote the median as $m$, we have:
$$P[X \geq m]=P[X \leq m]=0.50$$
For a discrete random variable, the mode is the value associated with the highest probability. As with population and sample data sets, the mode of a discrete random variable need not be unique.

统计代写|金融统计代写Mathematics with Statistics for Finance代考|Continuous Random Variables

We can also define the mean, median, and mode for a continuous random variable. To find the mean of a continuous random variable, we simply integrate the product of the variable and its probability density function (PDF). In the limit, this is equivalent to our approach to calculating the mean of a discrete random variable. For a continuous random variable, $X$, with a PDF, $f(x)$, the mean, $\mu$, is then:
$$\mu=\int_{x_{\min }}^{x_{\max }} x f(x) d x$$
The median of a continuous random variable is defined exactly as it is for a discrete random variable, such that there is a $50 \%$ probability that values are less than or equal to, or greater than or equal to, the median. If we define the median as $m$, then:
$$\int_{x_{\min }}^{m} f(x) d x=\int_{m}^{x_{\max }} f(x) d x=0.50$$
Alternatively, we can define the median in terms of the cumulative distribution function. Given the cumulative distribution function, $F(x)$, and the median, $m$, we have:
$$F(m)=0.50$$
The mode of a continuous random variable corresponds to the maximum of the density function. As before, the mode need not be unique.

统计代写|金融统计代写Mathematics with Statistics for Finance代考|Population and Sample Data

μ=1n∑一世=1n一种一世=1n(一种1+一种2+⋯+一种n−1+一种n)

μ^=1n∑一世=1nr一世=1n(r1+r2+⋯+rn−1+rn)

μ=∑一世=1np一世X一世

统计代写|金融统计代写Mathematics with Statistics for Finance代考|Continuous Random Variables

μ=∫X分钟X最大限度XF(X)dX

∫X分钟米F(X)dX=∫米X最大限度F(X)dX=0.50

F(米)=0.50

广义线性模型代考

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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。