Unit 1

Basic concepts in probability and simulation

This unit introduces probability models as a framework for reasoning about uncertainty in data. The primary goals are to (1) learn the mathematical language and notation needed to talk about statistical models (probability theory) (2) Learn how to translate mathematical models into Python code and vice versa. It will be important to go back and forth between mathematical notation and code, as this is something we do throughout the course! To this end, we will begin with some definitions and notation, and then get some practice computing things like marginal and conditional probabilities. This will be done both by hand (when possible) and from data.

Concepts

Probability models, random variables, sample spaces, probability distributions, Bernoulli random variables, joint marginal and conditional probabilities, independence, generating random variables in Python

Material:

• Notes and exercises
• Colab notebook from class (2024)
• Colab notebook from class (2025)
• Colab notebook for code examples in notes

Things to practice

• Notation for probability distributions, joint probabilities and conditional probabilities. Be comfortable with different notations.
• Identify the sample space for an experiment or random variable.
• Translate basic Python code to math and vice versa.
• Calculating conditional, marginal probabilities and checking whether two variables are independent:
  • by hand (given a table for discrete distribution)
  • or in Python (given samples)

Wikipedia References

• Mathematical model
• Random variable
• Sample space
• Bernoulli distribution
• Geometric distribution
• Binomial distribution
• Statistical independence