• Overview
• Details
• Exercises

Discrete probability studies outcomes on countable sample spaces (finite or countably infinite). Core ideas include probability mass functions, events, independence, and conditional probability.

• Sample space \(\Omega\), sigma-algebra \(\mathcal{F}\), probability measure \(\mathbb{P}\)
• Random variables and PMFs: \(p_X(x) = \mathbb{P}(X=x)\)
• Common distributions: Bernoulli, Binomial, Geometric, Poisson
• Linearity of expectation: \(\mathbb{E}[\sum X_i] = \sum \mathbb{E}[X_i]\) (no independence required)
• Variance and covariance; independence vs. uncorrelated
• Conditional probability and Bayes’ theorem
• Indicator variables for counting arguments

1. Compute the mean and variance of a Binomial\((n,p)\).
2. Show that the sum of independent Poisson variables is Poisson (add parameters).
3. Use indicator variables to derive the expected number of fixed points in a random permutation.