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统计代写|线性模型代写Linear Model代考|PSQF6270 Multivariate density functions

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统计代写|线性模型代写Linear Model代考|Multivariate density functions

In considering $n$ random variables $X_{1}, X_{2}, \ldots, X_{n}$, for which $x_{1}, x_{2}, \ldots$, $x_{n}$ represents a set of realized values we write the cumulative density function as
$$\operatorname{Pr}\left(X_{1} \leq x_{1}, X_{2} \leq x_{2}, \ldots, X_{n} \leq x_{n}\right)=F\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$
Then the density function is
$$f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\frac{\partial^{n}}{\partial x_{1} \partial x_{2} \ldots \partial x_{n}} F\left(x_{1}, x_{2}, \ldots, x_{n}\right) \text {. }$$
Conditions which $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ must satisfy are and
$$\begin{gathered} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) \geq 0 \text { for }-\infty<x_{i}<\infty \text { for all } i \ \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} d x_{2} \ldots d x_{n}=1 \end{gathered}$$

统计代写|线性模型代写Linear Model代考|Moments

The $k$ th moment about zero of the $i$ th variable is $E\left(x_{i}^{k}\right)$, the expected value of the $k$ th power of $x_{i}$ :
$$\mu_{x_{i}}^{(k)}=E\left(x_{i}^{k}\right)=\int_{-\infty}^{\infty} x_{i}^{k} g\left(x_{i}\right) d x_{i}$$
and on substituting from (7) for $g\left(x_{i}\right)$ this gives
$$\mu_{x_{i}}^{(k)}=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} x_{i}^{k} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} d x_{2} \ldots d x_{n} .$$
In particular, when $k=1$, the superscript $(k)$ is usually omitted and $\mu_{i}$ is written for $\mu_{i}^{(1)}$.
The covariance between the $i$ th and $j$ th variables for $i \neq j$ is
\begin{aligned} \sigma_{i j} &=E\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) \ &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) g\left(x_{i}, x_{j}\right) d x_{i} d x_{j} \ &=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n}, \end{aligned}
and similarly the variance of the $i$ th variable is
\begin{aligned} \sigma_{i i} \equiv \sigma_{i}^{2} &=E\left(x_{i}-\mu_{i}\right)^{2} \ &=\int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} g\left(x_{i}\right) d x_{i} \ &=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n} \end{aligned}

统计代写|线性模型代写Linear Model代考|Multivariate density functions

$$\operatorname{Pr}\left(X_{1} \leq x_{1}, X_{2} \leq x_{2}, \ldots, X_{n} \leq x_{n}\right)=F\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$

$$f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\frac{\partial^{n}}{\partial x_{1} \partial x_{2} \ldots \partial x_{n}} F\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$

$$f\left(x_{1}, x_{2}, \ldots, x_{n}\right) \geq 0 \text { for }-\infty<x_{i}<\infty \text { for all } i \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} d x_{2} \ldots d x_{n}=1$$

统计代写|线性模型代写Linear Model代考|Moments

$$\mu_{x_{i}}^{(k)}=E\left(x_{i}^{k}\right)=\int_{-\infty}^{\infty} x_{i}^{k} g\left(x_{i}\right) d x_{i}$$

$$\mu_{x_{i}}^{(k)}=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} x_{i}^{k} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} d x_{2} \ldots d x_{n}$$

$$\sigma_{i j}=E\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) \quad=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) g\left(x_{i}, x_{j}\right) d x_{i} d x_{j}=\int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n}$$

$$\sigma_{i i} \equiv \sigma_{i}^{2}=E\left(x_{i}-\mu_{i}\right)^{2} \quad=\int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} g\left(x_{i}\right) d x_{i}=\int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n}$$

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。