Probability and Statistics

Course Objective

After this course the student should be able to

• understand the setup of a probability space, events, probability and conditional probability (knowledge and insight);
• use the basic rules of probability theory (sum rule, product rule, law of total probability, Bayes' rule) (applying knowledge and insight);
• understand what a random variable is, what its distribution is, both for discrete and continuous random variables (this includes the concept of probability mass function, probability density function, and cumulative distribution function)

(knowledge and insight);

• compute the expectation and variance of a random variable (applying knowledge and insight);
• compute the expectation and variance of a sum of independent random variables (applying knowledge and insight);
• derive discrete probability distributions for probabilistic experiments associated to the Bernoulli, Binomial, Negative Binomial and Poisson distribution (applying knowledge and insight);
• compute probabilities for continuous random variables, specifically the uniform, exponential, and normal distribution (applying knowledge and insight);
• perform calculations based on the law of large numbers and central limit theorem (applying knowledge and insight);
• reproduce the definition of an estimator, carry out elementary computations involving point estimators (applying knowledge and insight);
• understand what confidence intervals are and be able to derive them in simple settings using a pivot (applying knowledge and insight);
• understand the setup of classical hypothesis testing (critical region, p-value, type of errors) and apply this to simple hypothesis testing problems (knowledge and insight);
• derive likelihood ratio tests in simple settings (applying knowledge and insight);

Course Content

From the book by From Pishro-Nik:

Chapter 1: Basic concepts
Chapter 3: Discrete random variables, Bernoulli, Binomial, Negative Binomial, Poisson
Chapter 4: Continuous random variables; sections 4.0, 4.1 and from 4.2 uniform, exponential and normal random variables
Chapter 7: Law of large numbers and central limit theorem; sections 7.0 and 7.1
Chapter 8: Classical statistics (point estimation, interval estimation, hypothesis testing); sections 8.1, 8.2, 8.3 and 8.4

Additional Information Teaching Methods

6 weeks with 4 hours lecturing each week. Exercise classes.

Attendance to lectures and classes is not mandatory but strongly recommended.

Method of Assessment

Midterm exam (50%) and final exam (50%) over all content. The resit exam is about all content.

Literature

• Hossein Pishro-Nik, Introduction to Probability, Statistics and Random Processes. The book can be ordered but is also freely available on www.probabilitycourse.com.

Additional Information Target Audience

Bachelor Artificial Intelligence (year 2)

Additional Information

In this course, generative AI (GenAI), such as ChatGPT or similar tools, may only be used for orientation and brainstorming, for example to explore ideas or clarify questions. GenAI may not be used to generate content that becomes part of the final submission. The final product must be based entirely on the student's own understanding, analysis, and wording. Incorrect use of GenAI is considered fraud. If established, the submitted work or assessment result will be declared invalid, and the Examination Board may impose further measures.

Recommended background knowledge

Elementary set theory
Calculus