MATH 333 概率论代写

Kolmogorov’s 0–1 Law

For any product probability $Q$ and $A \in \mathcal{B}^{(\infty)}$, $Q(A)=0$ or 1

Maximal Ergodic Lemma

For any positive linear contraction $U$ of $\mathcal{L}^{1}(X, \mathcal{A}, \mu)$, any $f \in \mathcal{L}^{1}(X, \mathcal{A}, \mu)$ and $n=0,1, \ldots$, let $A:=\left\{x: S_{n}^{+}(f)>\right.$ $0\} .$ Then $\int_{A} f d \mu \geq 0$

Proof .

Proof. For $r=0,1, \ldots, f+U S_{r}(f)=S_{r+1}(f)$. Note that since $U$ is positive and linear, $g \geq h$ implies $U g \geq U h$. Thus for $j=1, \ldots, n, S_{j}(f)=f+$ $U S_{j-1}(f) \leq f+U S_{n}^{+}(f)$

If $x \in A$, then $S_{n}^{+}(f)(x)=\max _{1 \leq j \leq n} S_{j}(f)(x)$. Combining gives for all $x \in A, f \geq S_{n}^{+}(f)-U S_{n}^{+}(f) .$ Since $S_{n}^{+}(f) \geq 0$ on $X$ and $S_{n}^{+}=0$ outside $A$, we have
$$
\begin{aligned}
\int_{A} f d \mu & \geq \int_{A}\left(S_{n}^{+}(f)-U S_{n}^{+}(f)\right) d \mu=\int_{X} S_{n}^{+}(f) d \mu-\int_{A} U S_{n}^{+}(f) d \mu \\
& \geq \int_{X} S_{n}^{+}(f) d \mu-\int_{X} U S_{n}^{+}(f) d \mu \geq 0
\end{aligned}
$$
since $U$ is a contraction.

Borel-Cantelli Lemma

 If $A_{n}$ are any events with $\sum_{n} P\left(A_{n}\right)<\infty$, then $P\left(\lim \sup A_{n}\right)=0 .$ If the $A_{n}$ are independent and $\sum_{n} P\left(A_{n}\right)=+\infty$, then $P\left(\lim \sup A_{n}\right)=1$

Proof .

The first part holds since for each $m$,
$$
P\left(\lim \sup A_{n}\right) \leq P\left(\bigcup_{n \geq m} A_{n}\right) \leq \sum_{n \geq m} P\left(A_{n}\right) \rightarrow 0 \quad \text { as } m \rightarrow \infty
$$

where $P\left(\bigcup_{n} A_{n}\right) \leq \sum_{n} P\left(A_{n}\right)$ (“Boole’s inequality”) If the $A_{n}$ are independent and $\sum_{n} P\left(A_{n}\right)=+\infty$, then for each $m$
$$
P\left(\Omega \backslash \bigcup_{n \geq m} A_{n}\right)=\Pi_{n \geq m}\left(1-P\left(A_{n}\right)\right)=0 \quad 
$$
Thus $P\left(\bigcup_{n \geq m} A_{n}\right)=1$ for all $m$. Let $m \rightarrow \infty$ to finish the proof.

The Bienaym´e-Chebyshev Inequality

For any real random variable $X$ and $t>0, P(|X| \geq t) \leq E X^{2} / t^{2}$

Proof .

$E X^{2} \geq E\left(X^{2} 1_{\{|X| \geq t\}}\right) \geq t^{2} P(|X| \geq t)$

MA2506代写概率代写统计代写认准uprivate™

E-mail: [email protected]  微信:shuxuejun


uprivate™是一个服务全球中国留学生的专业代写公司
专注提供稳定可靠的北美、澳洲、英国代写服务
专注于数学,统计,金融,经济,计算机科学,物理的作业代写服务

real analysis代写analysis 2, analysis 3请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。

抽象代数Galois理论代写

偏微分方程代写成功案例

代数数论代考

组合数学代考

统计作业代写

集合论数理逻辑代写案例

凸优化代写

统计exam代考

MATH 333: Probability and Statistics
Spring 2021 Coordinated Course Syllabus
NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences
takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. This means that
there must not be any forms of plagiarism, i.e., copying of homework, class projects, or lab assignments, or any
form of cheating in quizzes and exams. Under the University Code on Academic Integrity, students are obligated
to report any such activities to the Instructor.
DMS Online Exam Policy Spring 2021: Exams will be proctored using both Respondus LockDown
Browser+Monitor and Webex. Students will be required to join a Webex meeting from their phone with their
cameras on, and to access the exam through LockDown Browser on a Mac or Windows PC with webcam. Students
must follow all instructions related to environment checks and camera positioning.
Please be sure you read and fully understand our DMS Online Exam Policy.
COURSE INFORMATION
Course Description: Descriptive statistics and statistical inference. Topics include discrete and continuous
distributions of random variables, statistical inference for the mean and variance of populations, and graphical
analysis of data.
Number of Credits: 3
Prerequisites: MATH 112 with a grade of C or better or MATH 133 with a grade of C or better.
Course-Section and Instructors
Course-Section Instructor
Math 333-002 Professor S. Mahmood
Math 333-004 Professor P. Natarajan
Math 333-008 Professor D. Schmidt
Math 333-010 Professor K. Horwitz
Math 333-014 Professor K. Horwitz
Math 333-018 Professor W. Guo
Math 333-028 Professor K. Carfora
Math 333-102 Professor K. Carfora
Office Hours for All Math Instructors: Spring 2021 Office Hours and Emails