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数学代写|Gaussian random variables 数论代考

数学代写|Gaussian random variables 数论代考

数论代考

By definition, a random vector $X$ with values in $\mathrm{R}^{m}$ is called a (centered) gaussian vector if there exists a non-negative quadratic form $\mathrm{Q}$ on $\mathrm{R}^{m}$ such that the characteristic function $\varphi_{X}$ of $X$ is of the form
$$
\varphi_{\mathrm{X}}(t)=e^{-\mathrm{Q}(t) / 2}
$$
for $t \in \mathrm{R}^{m}$. The quadratic form can be recovered from $\mathrm{X}$ by the relation
$$
\mathrm{Q}\left(t_{1}, \ldots, t_{m}\right)=\sum_{1 \leqslant i, j \leqslant m} a_{i, j} t_{i} t_{j}
$$
with $a_{i, j}=\mathbf{E}\left(\mathrm{X}{i} \mathrm{X}{j}\right)$, and the (symmetric) matrix $\left(a_{i, j}\right){1 \leqslant i, j \leqslant m}$ is called the correlation matrix of $X$. The components $X{i}$ of $X$ are independent if and only if $a_{i, j}=0$ if $i \neq j$, i.e., if and only if the components of $X$ are orthogonal.
If $X$ is a gaussian random vector, then $X$ is mild, and in fact
$$
\sum_{\boldsymbol{k}} \mathrm{M}{m}(\mathrm{X}) \frac{t{1}^{k_{1}} \cdots t_{m}^{k_{m}}}{k_{1} ! \cdots k_{m} !}=\mathbf{E}\left(e^{t \cdot \mathrm{X}}\right)=e^{\mathrm{Q}(t) / 2}
$$
for $t \in \mathbf{R}^{m}$, so that the power series converges on all of $\mathrm{C}^{m}$. The Laplace transform $\psi_{\mathrm{X}}(z)=\mathbf{E}\left(e^{z \cdot \mathrm{X}}\right)$ is also defined for all $z \in \mathbf{C}^{m}$, and in fact
(B.9)
$$
\mathbf{E}\left(e^{z \cdot \mathrm{X}}\right)=e^{\mathrm{Q}(z) / 2} .
$$
For $m=1$, this means that a random variable is a centered gaussian if and only if there exists $\sigma \geqslant 0$ such that
$$
\varphi_{\mathrm{X}}(t)=e^{-\sigma^{2} t / 2}
$$
and in fact we have
$$
\mathbf{E}\left(\mathrm{X}^{2}\right)=\mathbf{V}(\mathrm{X})=\sigma^{2}
$$
If $\sigma=1$, then we say that $\mathrm{X}$ is a standard gaussian random variable (also sometimes called a standard normal random variable). We then have
$$
\mathbf{P}(a<\mathrm{X}<b)=\frac{1}{\sqrt{2 \pi}} \int_{a}^{b} e^{-x^{2} / 2} d x
$$
for all real numbers $a<b$.
EXERCISE B.7.1. We recall a standard proof of the fact that the measure on $\mathbf{R}$ given by
$$
\mu=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} d x
$$
is indeed a gaussian probability measure with variance $1 .$
(1) Define
$$
\varphi(t)=\varphi_{\mu}(t)=\frac{1}{\sqrt{2 \pi}} \int_{\mathbf{R}} e^{i t x-x^{2} / 2} d x
$$
for $t \in \mathbf{R}$. Prove that $\varphi$ is of class $C^{1}$ on $\mathbf{R}$ and satisfies $\varphi^{\prime}(t)=-t \varphi(t)$ for all $t \in \mathbf{R}$ and $\varphi(0)=1$.
(2) Deduce that $\varphi(t)=e^{-t^{2} / 2}$ for all $t \in \mathbf{R}$. [Hint: This is an elementary argument with ordinary differential equations, but because the order is 1 , one can define $g(t)=$ $e^{t^{2} / 2} \varphi(t)$ and check by differentiation that $g^{\prime}(t)=0$ for all $t \in \mathbf{R}$.]
148
We will use the following simple version of the Central Limit Theorem:
THEOREM B.7.2. Let $\mathrm{B} \geqslant 0$ be a fixed real number. Let $\left(\mathrm{X}{n}\right)$ be a sequence of independent real-valued random variables with $\left|\mathrm{X}{n}\right| \leqslant \mathrm{B}$ for all $n$. Let
$$
\alpha_{n}=\mathrm{E}\left(\mathrm{X}{n}\right), \quad \beta{n}=\mathrm{V}\left(\mathrm{X}{n}^{2}\right) $$ Let $\sigma{\mathrm{N}} \geqslant 0$ be defined by
$$
\sigma_{\mathrm{N}}^{2}=\beta_{1}+\cdots+\beta_{\mathrm{N}}
$$
for $\mathrm{N} \geqslant 1$. If $\sigma_{\mathrm{N}} \rightarrow+\infty$ as $n \rightarrow+\infty$, then the random variables
$$
\mathrm{Y}{\mathrm{N}}=\frac{\left(\mathrm{X}{1}-\alpha_{1}\right)+\cdots+\left(\mathrm{X}{\mathrm{N}}-\alpha{\mathrm{N}}\right)}{\sigma_{\mathrm{N}}}
$$
converge in law to a standard gaussian random variable.
Proof. Although this is a very simple case of the general Central Limit Theorem for sums of independent random variables (indeed, even of Lyapunov’s well-known version), we give a proof using Lévy’s criterion for convenience. First of all, we may assume that $\alpha_{n}=0$ for all $n$ by replacing $\mathrm{X}{n}$ by $\mathrm{X}{n}-\alpha_{n}$ (up to replacing B by $2 \mathrm{~B}$, since $\left|\alpha_{n}\right| \leqslant \mathrm{B}$ ). by

By definition, a random vector $X$ with values in $\mathrm{R}^{m}$ is called a (centered) gaussian vector if there exists a non-negative quadratic form $\mathrm{Q}$ on $\mathrm{R}^{m}$ such that the characteristic function $\varphi_{X}$ of $X$ is of the form
$$
\varphi_{\mathrm{X}}(t)=e^{-\mathrm{Q}(t) / 2}
$$
for $t \in \mathrm{R}^{m}$. The quadratic form can be recovered from $\mathrm{X}$ by the relation
$$
\mathrm{Q}\left(t_{1}, \ldots, t_{m}\right)=\sum_{1 \leqslant i, j \leqslant m} a_{i, j} t_{i} t_{j}
$$
with $a_{i, j}=\mathbf{E}\left(\mathrm{X}{i} \mathrm{X}{j}\right)$, and the (symmetric) matrix $\left(a_{i, j}\right){1 \leqslant i, j \leqslant m}$ is called the correlation matrix of $X$. The components $X{i}$ of $X$ are independent if and only if $a_{i, j}=0$ if $i \neq j$, i.e., if and only if the components of $X$ are orthogonal.
If $X$ is a gaussian random vector, then $X$ is mild, and in fact
$$
\sum_{\boldsymbol{k}} \mathrm{M}{m}(\mathrm{X}) \frac{t{1}^{k_{1}} \cdots t_{m}^{k_{m}}}{k_{1} ! \cdots k_{m} !}=\mathbf{E}\left(e^{t \cdot \mathrm{X}}\right)=e^{\mathrm{Q}(t) / 2}
$$
for $t \in \mathbf{R}^{m}$, so that the power series converges on all of $\mathrm{C}^{m}$. The Laplace transform $\psi_{\mathrm{X}}(z)=\mathbf{E}\left(e^{z \cdot \mathrm{X}}\right)$ is also defined for all $z \in \mathbf{C}^{m}$, and in fact
(B.9)
$$
\mathbf{E}\left(e^{z \cdot \mathrm{X}}\right)=e^{\mathrm{Q}(z) / 2} .
$$
For $m=1$, this means that a random variable is a centered gaussian if and only if there exists $\sigma \geqslant 0$ such that
$$
\varphi_{\mathrm{X}}(t)=e^{-\sigma^{2} t / 2}
$$
and in fact we have
$$
\mathbf{E}\left(\mathrm{X}^{2}\right)=\mathbf{V}(\mathrm{X})=\sigma^{2}
$$
If $\sigma=1$, then we say that $\mathrm{X}$ is a standard gaussian random variable (also sometimes called a standard normal random variable). We then have
$$
\mathbf{P}(a<\mathrm{X}<b)=\frac{1}{\sqrt{2 \pi}} \int_{a}^{b} e^{-x^{2} / 2} d x
$$
for all real numbers $a<b$.
EXERCISE B.7.1. We recall a standard proof of the fact that the measure on $\mathbf{R}$ given by
$$
\mu=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} d x
$$
is indeed a gaussian probability measure with variance $1 .$
(1) Define
$$
\varphi(t)=\varphi_{\mu}(t)=\frac{1}{\sqrt{2 \pi}} \int_{\mathbf{R}} e^{i t x-x^{2} / 2} d x
$$
for $t \in \mathbf{R}$. Prove that $\varphi$ is of class $C^{1}$ on $\mathbf{R}$ and satisfies $\varphi^{\prime}(t)=-t \varphi(t)$ for all $t \in \mathbf{R}$ and $\varphi(0)=1$.
(2) Deduce that $\varphi(t)=e^{-t^{2} / 2}$ for all $t \in \mathbf{R}$. [Hint: This is an elementary argument with ordinary differential equations, but because the order is 1 , one can define $g(t)=$ $e^{t^{2} / 2} \varphi(t)$ and check by differentiation that $g^{\prime}(t)=0$ for all $t \in \mathbf{R}$.]
148
We will use the following simple version of the Central Limit Theorem:
THEOREM B.7.2. Let $\mathrm{B} \geqslant 0$ be a fixed real number. Let $\left(\mathrm{X}{n}\right)$ be a sequence of independent real-valued random variables with $\left|\mathrm{X}{n}\right| \leqslant \mathrm{B}$ for all $n$. Let
$$
\alpha_{n}=\mathrm{E}\left(\mathrm{X}{n}\right), \quad \beta{n}=\mathrm{V}\left(\mathrm{X}{n}^{2}\right) $$ Let $\sigma{\mathrm{N}} \geqslant 0$ be defined by
$$
\sigma_{\mathrm{N}}^{2}=\beta_{1}+\cdots+\beta_{\mathrm{N}}
$$
for $\mathrm{N} \geqslant 1$. If $\sigma_{\mathrm{N}} \rightarrow+\infty$ as $n \rightarrow+\infty$, then the random variables
$$
\mathrm{Y}{\mathrm{N}}=\frac{\left(\mathrm{X}{1}-\alpha_{1}\right)+\cdots+\left(\mathrm{X}{\mathrm{N}}-\alpha{\mathrm{N}}\right)}{\sigma_{\mathrm{N}}}
$$
converge in law to a standard gaussian random variable.
Proof. Although this is a very simple case of the general Central Limit Theorem for sums of independent random variables (indeed, even of Lyapunov’s well-known version), we give a proof using Lévy’s criterion for convenience. First of all, we may assume that $\alpha_{n}=0$ for all $n$ by replacing $\mathrm{X}{n}$ by $\mathrm{X}{n}-\alpha_{n}$ (up to replacing B by $2 \mathrm{~B}$, since $\left|\alpha_{n}\right| \leqslant \mathrm{B}$ ). by

数论代写

数论是纯粹数学的分支之一,主要研究整数的性质。整数可以是方程式的解(丢番图方程)。有些解析函数(像黎曼ζ函数)中包括了一些整数、质数的性质,透过这些函数也可以了解一些数论的问题。透过数论也可以建立实数和有理数之间的关系,并且用有理数来逼近实数(丢番图逼近)。
按研究方法来看,数论大致可分为初等数论和高等数论。初等数论是用初等方法研究的数论,它的研究方法本质上说,就是利用整数环的整除性质,主要包括整除理论、同余理论、连分数理论。高等数论则包括了更为深刻的数学研究工具。它大致包括代数数论、解析数论、计算数论等等。

数学代写代考| Discrete Mathematics 离散数学

代写数论

其他相关科目课程代写:组合学Combinatorics集合论Set Theory概率论Probability组合生物学Combinatorial Biology组合化学Combinatorial Chemistry组合数据分析Combinatorial Data Analysis

my-assignmentexpert愿做同学们坚强的后盾,助同学们顺利完成学业,同学们如果在学业上遇到任何问题,请联系my-assignmentexpert™,我们随时为您服务!

在中世纪时,除了1175年至1200年住在北非和君士坦丁堡斐波那契有关等差数列的研究外,西欧在数论上没有什么进展。

数论中期主要指15-16世纪到19世纪,是由费马梅森欧拉高斯勒让德黎曼希尔伯特等人发展的。最早的发展是在文艺复兴的末期,对于古希腊著作的重新研究。主要的成因是因为丢番图的《算术》(Arithmetica)一书的校正及翻译为拉丁文,早在1575年Xylander曾试图翻译,但不成功,后来才由Bachet在1621年翻译完成。

计量经济学代考

计量经济学是以一定的经济理论和统计资料为基础,运用数学、统计方法与电脑技术,以建立经济计量模型为主要手段,定量分析研究具有随机性特性的经济变量关系的一门经济学学科。 主要内容包括理论计量经济学和应用经济计量学。 理论经济计量学主要研究如何运用、改造和发展数理统计的方法,使之成为经济关系测定的特殊方法。

相对论代考

相对论(英語:Theory of relativity)是关于时空和引力的理论,主要由愛因斯坦创立,依其研究对象的不同可分为狭义相对论和广义相对论。 相对论和量子力学的提出给物理学带来了革命性的变化,它们共同奠定了现代物理学的基础。

编码理论代写

编码理论(英语:Coding theory)是研究编码的性质以及它们在具体应用中的性能的理论。编码用于数据压缩加密纠错,最近也用于网络编码中。不同学科(如信息论电机工程学数学语言学以及计算机科学)都研究编码是为了设计出高效、可靠的数据传输方法。这通常需要去除冗余并校正(或检测)数据传输中的错误。

编码共分四类:[1]

  1. 数据压缩(或信源编码
  2. 前向错误更正(或信道编码
  3. 加密编码
  4. 线路码

数据压缩和前向错误更正可以一起考虑

复分析代考

学习易分析也已经很冬年了,七七八人的也续了圧少的书籍和论文。略作总结工作,方便后来人学 Đ参考。
复分析是一门历史悠久的学科,主要是研究解析函数,亚纯函数在复球面的性质。下面一昭这 些基本内容。
(1) 提到复变函数 ,首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根, 极坐标与 $x y$ 坐标的转换,复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题,此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面,从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式,这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后,就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中, 定义几乎是一致的。在引入了闭曲线和曲线积分之后,就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。

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