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. 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 $P\left(\lim \sup A_{n}\right)=1$
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 $t>0, P(|X| \geq t) \leq E X^{2} / t^{2}$
$E X^{2} \geq E\left(X^{2} 1_{\{|X| \geq t\}}\right) \geq t^{2} P(|X| \geq t)$
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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
