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MATP4600 probability theory课程简介
STAT 414 is an introductory course in probability theory, which covers the fundamental concepts, theorems, and applications of probability. The course aims to provide students with a solid foundation in probability theory, including both theoretical and practical aspects of the subject.
The three main goals of the course are:
- To learn the theorems of basic probability: The course covers the fundamental theorems of probability, including the axioms of probability, conditional probability, Bayes’ theorem, independence, random variables, and probability distributions.
- To learn applications and methods of basic probability: The course also covers various applications of probability theory, including counting techniques, discrete and continuous probability distributions, and limit theorems. Students will learn how to use these methods to solve real-world problems.
- To develop theoretical problem-solving skills: The course emphasizes the development of theoretical problem-solving skills, including the ability to understand and use mathematical proofs, and to reason logically and rigorously about probability concepts. Students will be challenged to solve theoretical problems and to apply their understanding of probability to solve practical problems.
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PREREQUISITES
The three main goals of the course are:
- To learn the theorems of basic probability: The course covers the fundamental theorems of probability, including the axioms of probability, conditional probability, Bayes’ theorem, independence, random variables, and probability distributions.
- To learn applications and methods of basic probability: The course also covers various applications of probability theory, including counting techniques, discrete and continuous probability distributions, and limit theorems. Students will learn how to use these methods to solve real-world problems.
- To develop theoretical problem-solving skills: The course emphasizes the development of theoretical problem-solving skills, including the ability to understand and use mathematical proofs, and to reason logically and rigorously about probability concepts. Students will be challenged to solve theoretical problems and to apply their understanding of probability to solve practical problems.
MATP4600 probability theory HELP（EXAM HELP， ONLINE TUTOR）
Theorem 4.6.10. Chebychev’s inequality. Let $X \in L$ be arbitrary. Then the following conditions hold:
- (First and common version of Chebychev’s inequality.) If $t>0$ is a regular point of the integrable function $|X|$, then we have $\mu(|X|>t) \leq t^{-1} I|X|$.(Second version.) If $I|X|0$, then for each $s>0$, we have $(|X|>s) \subset B$ for some integrable set $B$ with $\mu(B)0$ is given without any assurance that the set $(|X|>s)$ is integrable.
Proof. 1. $1_{(|X|>t)} \leq t^{-1}|X|$. Assertion 1 follows.
- Take an arbitrary regular point $t$ of the integrable function $X$ in the open interval $\left(b^{-1} I|X| s, s\right)$. Let $B \equiv(|X|>t)$. By Assertion 1, we then have $\mu(B) \leq t^{-1} I|X|s) \subset(|X|>t) \equiv B$. Assertion 2 is proved.
Proposition 4.6.12. The product of a bounded continuous function of an integrable function and an integrable indicator is integrable. Suppose $X \in L$, $A$ is an integrable set, and $f \in C_{u b}(R)$. Then $f(X) 1_A \in L$. In particular, if $X \in L$ is bounded, then $X 1_A$ is integrable.
Proof. 1 By Assertion 3 of Corollary 4.6.9, all but countably many real numbers are regular points of $X$ relative to $A$. In other words, there exists a countable subset $J$ of $R$ such that each $t \in J_c$ is a regular point of the integrable function $X$ relative to the integrable set $A$. Here $J_c$ denotes the metric complement of the set $J$ in $R$.
- Let $c>0$ be so large that $|f| \leq c$ on $R$. Let $\delta_f$ be a modulus of continuity of the function $f$. Let $\varepsilon>0$ be arbitrary. Since $X$ is integrable, there exists $a>0$ so large that
$$
I|X|-I|X| \wedge(a-1)<\varepsilon
$$
Since $f$ is uniformly continuous, there exists a sequence $-a=t_0<t_1<\cdots<$ $t_n=a$ such that $\bigvee_{i=1}^n\left(t_i-t_{i-1}\right)<\delta_f(\varepsilon)$. Then
$$
Y \equiv \sum_{i=1}^n f\left(t_i\right) 1_{(t(i-1)<X \leq t(i)) A}
$$ is an integrable function, where we recall the convention established in Definition 4.6.11 regarding the choice of regular points. Moreover, since $1_{(|X|>a) A} \leq|X|-$ $|X| \wedge(a-1)$, we have - $$
- \begin{aligned}
- \left|f(X) 1_A-Y\right| & \equiv\left|f(X) 1_A-\sum_{i=1}^n f\left(t_i\right) 1_{(t(i-1)a) A} \
- & \leq\left|\sum_{i=1}^n\left(f(X)-f\left(t_i\right)\right) 1_{(t(i-1)a) A} \
- & \leq \varepsilon 1_A+c(|X|-|X| \wedge(a-1)),
- \end{aligned}
- $$
- where
- $$
- \begin{aligned}
- & I\left(\varepsilon 1_A+c(|X|-|X| \wedge(a-1))\right) \
- & \quad=\varepsilon \mu(A)+c(I|X|-I|X| \wedge(a-1))<\varepsilon \mu(A)+c \varepsilon \rightarrow 0
- \end{aligned}
- $$
- as $\varepsilon \rightarrow 0$. Hence, by Theorem 4.5.10, $f(X) 1_A \in L$. This proves the first part of the proposition.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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