Course Syllabus

Math 5010 / Math 6805 introduction to probability is a 3 credit course. The prerequisite is "C" or better in Math 2210 or Math 1260 or Math 1280 or Math 1321 or Math 3140.

The course aims to develop the ability to understand and describe probabilistic phenomena in a mathematical fashion, and the ability to apply mathematical tools to derive useful facts about probabilistic phenomena.

Upon completing this course, you'll be able to do the following. 

• Understand the notion of random variables, use them to describe random phenomena, express a more complicated random variable by simpler ones.
• Understand the notion of events, use them to describe random phenomena, be familiar with the basic properties of the probabilities of events and apply them.
• Apply combinatorics to calculate the probabilities in sampling with and without replacements and related problems.
• Understand the notion of conditional probabilities, understand Bayes' formula and apply it, find the conditional probabilities in (certain) concrete problems.
• Understand the notion of independence, decide whether certain given events or random variables are independent in (certain) concrete problems.
• Be familiar with the iid Bernoulli sequence, understand how it gives rise to the geometric and binomial random variables, and be familiar with these random variables.
• Understand the notion of a continuous random variable and of its Probability Distribution Function (PDF), express the probabilities related to such a random variable in terms of its PDF, evaluate such probabilities in (certain) concrete instances.
• Understand the notion of a uniform random variable, understand the ambiguity behind the colloquial term "choosing something at random/randomly".
• Be familiar with the cumulative distribution function, expectation, variance, moments, and moment generating function of a random variable, calculate these functions/quantities in (certain) concrete instances, understand how moment generating function generates moments, be familiar with and can apply the "expansion techniques" in calculating moments.
• Understand the notion of a sample space, understand its relation with a random variable.
• Understand the statement of the Law of Large Numbers, understand how it applies in Monte Carlo simulations, estimate the errors in those simulations.
• Understand the statement of the Central Limit Theorem (CLT), be familiar with the Gaussian random variable and its properties, apply the CLT to estimate the confidence level.
• Be familiar with the Poisson process and Poisson distribution and their properties, understand how Bernoulli trials give rise to the Poisson process.
• Be familiar with the joint distribution, marginal distribution, and conditional distribution of a set of random variables, find these distributions in (certain) concrete instances.
• Be familiar with the notion of independence in the context of joint distributions, be familiar with the covariance and correlation of a set of random variables, find these quantities in (certain) concrete instances.
• Understand the notion of a conditional expectation as a random variable, find such conditional expectations in (certain) concrete instances, understand and be familiar with the tower property of conditional expectations, apply the property in (certain) concrete instances.

The course includes the following components.

• Lectures and in-class problems
• Quizzes
• Exams

Your performance will be evaluated as follows.

• Quizzes 40% lowest one dropped
• Exams 60% all 4 taken and equally weighted

Out of 100,

• a final score of 92.00 or above will receive the letter grade A,
• a final score from 88.00 to 91.99 will receive the letter grade A−,
• a final score from 84.00 to 87.99 will receive the letter grade B+,
• a final score from 80.00 to 83.99 will receive the letter grade B,
• a final score from 76.00 to 79.99 will receive the letter grade B−,
• a final score from 71.00 to 75.99 will receive the letter grade C+,
• a final score from 66.00 to 71.99 will receive the letter grade C,
• a final score from 61.00 to 65.99 will receive the letter grade C−,
• a final score from 56.00 to 60.99 will receive the letter grade D+,
• a final score from 51.00 to 55.99 will receive the letter grade D,
• a final score from 46.00 to 50.99 will receive the letter grade D−.

We will not round the scores.

Classes will meet in LCB 215 (LeRoy Cowles Building) 6-7:30 pm on Monday and Wednesday.

If you feel ill, please stay home and take some rest. If you miss a quiz or exam due to illness, please let me know (licheng.tsai@utah.edu) and we'll find ways around it.

I'll hold an office hour 1:20-1:50 on Monday and 3:30-4:00 on Wednesday in JWB 322 (John Widtsoe Building). Come to my office hours for help or questions. If the time doesn't work for you, we can try to schedule an appointment; you can also email me (licheng.tsai@utah.edu) your questions.

Lecture notes. This is the primary source of study and reference for this course. To access them, click "Files" in the course navigation menu.

If any conflict may prevent you from completing or attending any assignment (quiz, exam) on time, you should notify me by email as soon as you become aware of the conflict, even if it's just a possibility. If it is an emergency, you should notify me by email as soon as it is safe for you to do so. I may not be able to accommodate requests that do not follow the above guidelines.

I'd also like to let you be aware of the University's resources for students' well being. To find out about them, scroll down to "University Policies".

In-class problems

There'll be a few problems in the lecture notes named "In-class problems" which I'll ask you to work on in-class.

The purpose of these problems is to help you absorb the material. Your work on these in-class problems will not contribute to your final score, and you need not worry about getting the full or correct solution on the spot. During the class, you should try these problems, see if there's any impediments, confusions, etc., and ask me about them. The more questions the better.

To review these problems after a meeting, click "File" in the course navigation menu and go to the lecture notes you want. We will not post the answer or solution key to these in-class problems, and you can ask me about them if you have any questions.

Quizzes

When and where. There'll be a quiz in most Monday classes, and also Wednesday 1/22 and 2/19. See "Schedule" for the dates. They will be given in-class, with a window of 10-minute long.

What. Each quiz will cover the lecture notes of the week(s) in the numbering of the quizz. For example,

• "Quiz 01" will cover lecture notes "Week 01".
• "Quiz 09-10" will cover lecture notes "Week 09" and "Week 10".

Most of the quiz problems will be similar to the "In-class problems" in the lecture notes. The best way to prepare for the quizzes is to review the lecture notes and those "In-class problems". The first problem in every quiz will be "I have studied for this quizz. Check the box", which worths a certain number of points. Make sure to study for the quiz and check that box.

Policies. The quizzes are closed-book and will be given in the written form.

During the quiz window, do not use any electronic devices (including math software and calculators), do not refer to any texts or books or notes, and do not seek information from anyone other than me (the instructor). If you have any questions during the quiz, please ask me.

If you have difficulties coming to a quiz (for example not able to attend a class), please let me know by email at your earliest convenience.

Weights. The one lowest score will be dropped, and the rest will be equally weighted. Altogether they contribute to 40% of your final score.

To help you become familiar with the material and also study for the exams, I've prepare some practice problem sets. To access them, click the "Files" tab on the left. Set 1-4 corresponds to the material of Exam 1-4 respectively. These problems are optional and will not be graded. You are of course welcome to ask me any questions about them.

Not for quizzes! These problem sets are not indented to be the primary resources for preparing for the quizzes. For quizzes, the in-class problems and lecture notes are the best resources.

When and where. There'll be 4 exams. See "Schedule" for the dates. They will be given in-class, with an exam window of 90-minute long.

What. Each exam will cover roughly 1/4 of materials of the semester. All exams are non-cumulative, in the sense that any exam will not directly cover the materials of previous exams. Though, materials later in the semester may be built on materials earlier in the semester. The materials of each exam will be announced in-class and on Canvas when the time gets closer.

Policies. The exams are closed-book and will be given in the written form. 

During the exam window, do not use any electronic devices (including math software and calculators), do not refer to any texts or books or notes, and do not seek information from anyone other than me (the instructor). If you have any questions during the exam, please ask me.

If you have difficulties coming to an exam (for example not able to attend a class), please let me know by email at your earliest convenience.

Weights. Each exam weighs 15% of your final score. Altogether they contribute to 60% of your final score.