# Lecture Notes Probability Theory And Random Processes Homework

**Fall 2015: Aug 24 - Dec 11**

** Instructor: ** Jia Li

417A Thomas Building, phone: 814-863-3074, email: jiali@stat.psu.edu

office hours: Mon 2:15pm-3:00pm, Wed 2:15-3:00pm, or by appointment

** Teaching assistant: ** Gregory Bopp

Office: 333 Thomas Building, Email: gxb951@psu.edu

Office hours: Tuesday 11am-12pm, & Thursday 11am-12pm

**Lectures: ** MWF 1:25-2:15pm 216 Thomas

**Course homepage: ** http://www.personal.psu.edu/jol2/course/stat416 .

** Description of the course: **

Introduction to the elementary theory of stochastic processes. The course will be focused on conditional probability and conditional expectation, Markov chains, the Poisson process and its variations, continuous-time Markov chain including birth and death processes. These topics are covered by Chapter 3 to 6 in the text book. We will briefly review elementary probability, which corresponds to Chapter 1 and 2 in the text book, at the beginning of the course.

Prerequisites: Math 230 (calculus) and Stat 414, or Stat 318 (elementary probability). A fair amount of mathematical expertise (analytical thinking and proof-writing) is expected.

**Textbooks:**

Required: *Introduction to Probability Models, 11th ed.*, by Sheldon Ross

** Grading: **

- Homeworks: 10%
- Two midterms: 40%
- Final exam: 50%

**Exams:**

- Midterm 1: Oct 2 (Friday) & Midterm 2: Nov 6 (Friday)
- Final: TBD

- Dates are subject to potential changes, but will be announced well before the exams.
- Exams will all be closed book.
- For midterm, you can bring a
*one-sided*fact sheet no larger than 8.5x11 inches; for final, a*two-sided*fact sheet no larger than 8.5x11 inches is allowed. - Makeup exams are permitted only for very exceptional cases; and verifiable reasons are required for such exceptions. Students who need makeup exams should plan early and let me know at least a week ahead of time. A written notice is required.

**Homeworks:**

- Starting from the second week, homework will be assigned almost every week on Wednesday and is due on Wednesday in the following week, unless specially noticed.
- Homework is required to be submitted at the beginning of the class on Wednesday.
- Late homework will not be accepted, but one homework with the lowest score will not be included in the final evaluation.
- Discussion on homework is encouraged. However, each student must turn in his/her own written work that reflects his/her own understanding of the material. It is a violation of course policy to copy solutions from others, textbooks, Websites, or previous instances of this course.

**Topics**

- Introduction to probability theory (Chapter 1), Week 1-2
- Random variables (Chapter 2), Week 3-5
- Conditional probability and conditional expectation (Chapter 3), Week 6-8
- Discrete-time Markov chain (Chapter 4), Week 9-13
- Exponential distribution and the Poisson process (Chapter 5), Week 13-15
- Continuous-time Markov chain, birth and death processes (Chapter 6), Week 15-16

**Academic Integrity:**

All Penn State and Eberly College of Science policies regarding academic integrity apply to this course. See http://science.psu.edu/current-students/Integrity/Policy.html for details.

**Lecture Dates**, academic calendar

**Homework**

**Lecture notes, homework hints, and solutions**

* ------------- Updated on August, 2015 --------------- *

36-217: Probability Theory and Random ProcessesCourse DescriptionThe theory of probability and random processes provides the mathematical tools needed to model uncertainty in virtually all scientiFc areas. Nowadays, probabilistic models are key for the development and analysis of scientiFc theories and of many algorithms, with countless applications, including the analysis of network dynamics of circuit failure rates, the development of algorithms for computer vision, machine learning, image processing, cryptography, system performance assessment, business inventory, marketing, Fnance, medicine, etc.This is a Frst course in probability theory that is designed to prepare you to develop and analyze basic probabilistic models for describing and studying uncertain or random phenomena and to make better predictions and decisions. By the end of this course, you will1.Possess an adequate background and understanding of basic concepts in probability theory;2.Be able to apply probability terminology and formalism correctly to represent elementary random experiments and quantify uncertainty;3.Be proFcient in the calculus-based mathematical skills needed to solve problems in basic probability.Course Objectives1.Basic Probability.•Describe the sample space of an experiment using set notation.•±ind the probabilities of complements, unions, and intersections of events.•Use counting tools to enumerate the number of equally likely outcomes of simple experiments.•Use the law of total probability and Bayes’ Rule to calculate probabilities.•DeFne and identify independence of events.•Calculate conditional probabilities of events.2.Random Variables•Describe the random variable associated with a question of interest.•Describe and identify discrete and continuous random variables.

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