assignmentutor-lab™ 为您的留学生涯保驾护航 在代写概率论Probability theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写概率论Probability theory代写方面经验极为丰富，各种代写概率论Probability theory相关的作业也就用不着说。

• 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

数学代写|概率论代写Probability theory代考|CHARACTERISTIC FUNCTIONS AND THEIR PROPERTIES

In the previous section we introduced the reader to the generating functions. They make it possible to simplify the solution of a number of probabilistic problems. In particular, the generating functions help relatively easily obtain various moment characteristics of random variables which will be used in the future when studying variances and other moments. However, it should be noted that these functions have some limited use and are used for a very limited albeit quite important set of discrete random variables. The main convenient properties of generating functions have inherited the socalled characteristic functions which are already defined for any probability distributions. We note that almost always the relations obtained for the characteristic functions remain valid (as amended) for generating functions. Therefore, we restrict ourselves to a more detailed study of the useful properties of the characteristic functions.

Consider an arbitrary random variable $\xi$ having some distribution function $\mathrm{F}(x)$. The characteristic function $\psi(t)$ of this random variable is given by
$$\psi(t)=E\left(e^{i t \xi}\right)=\int_{-\infty}^{\infty} e^{i t x} \mathrm{dF}(\mathrm{x}) .-\infty<t<\infty .$$

For discrete random variable $\xi$ taking values $x_k,=1,2, \ldots$, with the corresponding probabilities $p_k=P\left{\xi=x_k\right}$ definition (3.2.1) can be rewritten in the form
$$\psi(t)=\sum_k \exp \left(i t x_k\right) p_k .$$
Equality (3.2.2) allows for random variables taking only integer nonnegative values, i.e., having some generating function $P(s)$ as follows connect these two related functions:
$$\psi(t)=P(\exp (i t))$$
If $\xi$ has some distribution density $f(x)$ then we obtain that
$$\psi(t)=\int_{-\infty}^{\infty} \exp (i t x) f(x) d x$$
The reader may be confused by the fact that introducing random variables, we defined them as some measurable functions that take real values, and when considering a new object, we are already dealing with complex-valued functions $\exp (i t \xi)$. In this case, it would be possible to specifically introduce such an object as complex-valued random variables but it is easier giving a definition of a characteristic function to give its alternative notation in the form
$$\psi(t)=E \cos (t \xi)+i E \sin (t \xi),$$
where the real and imaginary parts of $f(t)$ are already presented in the form of mathematical expectations of completely “legitimate” (real-valued for each fixed value of the parameter $t$ ) random variables $\cos (t \xi)$ and $\sin (t \xi)$.
Here are a few properties of characteristic functions that often make it easier to work with random variables and their moment characteristics.

数学代写|概率论代写Probability theory代考|Characteristic Functions and Polya Criterion

The characteristic functions of those distributions that can be attributed to the category of classical and more often used others were given above. A number of operations can significantly expand the set of characteristic functions.

It was already noted in (3.2.10) that if $\eta=-\xi$, then $\psi_{\xi}(\mathrm{t})=\psi_{\xi}(-\mathrm{t})=$ $\overline{\psi_{\xi}(t)}$

Often, in order not to work with complex-valued functions, the so-called symmetrization operation is used. In this case, instead of the initial random variable $\xi$, the difference $\mu=\xi-\eta$ is considered, in which the quantities $\xi$ and $\eta$ are independent and have the same distribution. Even if the corresponding characteristic function $\psi(t)$ of these variables was complex, then the characteristic function of the difference $\mu=\xi-\eta$ has the form $\psi(t) \psi(-t)=|\psi(t)|^2$, i.e., becomes real-valued. Having performed a series of operations with the sums of the thus symmetrized random terms and obtained some results for such sums, we can transfer these results for the sums of the initial (considered before the symmetrization operation) random variables.

We note one more possibility to significantly increase the set of functions that are characteristic.

Let us deal with a set of distribution functions $f_k(x), k=1,2, \ldots$. It is easy to verify that any mixture of these functions with non-negative weights $p_1, p_2, \ldots$, the sum of which is unity, i.e., function
$$F(x)=\sum_k p_k F_k(x)$$
is a distribution function with the corresponding characteristic functions $\psi_k(t), k=1,2, \ldots$, we obtain a similar result for them, i.e., any linear combination $$\psi(t)=\sum_k p_k \psi(t)$$
with the above non-negative weights is a characteristic function. This fact and the desire to significantly expand the set of real characteristic functions made it possible to describe a fairly rich set of such functions. This is the socalled Polya criterion. The following result holds.

概率论代考

数学代写|概率论代写Probability theory代考|CHARACTERISTIC FUNCTIONS AND THEIR PROPERTIES

$$\psi(t)=E\left(e^{i t \xi}\right)=\int_{-\infty}^{\infty} e^{i t x} \mathrm{dF}(\mathrm{x}) .-\infty<t<\infty .$$

$$\psi(t)=\sum_k \exp \left(i t x_k\right) p_k .$$

$$\psi(t)=P(\exp (i t))$$

$$\psi(t)=\int_{-\infty}^{\infty} \exp (i t x) f(x) d x$$

$$\psi(t)=E \cos (t \xi)+i E \sin (t \xi),$$

数学代写|概率论代写Probability theory代考|Characteristic Functions and Polya Criterion

$$F(x)=\sum_k p_k F_k(x)$$

$$\psi(t)=\sum_k p_k \psi(t)$$

有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师