Probability
Probability spaces, expectation, conditional expectation and the four modes of convergence, stated precisely.
Level 0 — Foundations
Level 1 — Core
- Expectation, Variance and Conditional Expectation10 min
Expectation as an integral, the variance identities used in every bias–variance computation, and conditional expectation as an orthogonal projection.
Tail formula for the expectationBias–variance and variance decompositionsConditional expectation as an L2 projection - Modes of Convergence, LLN and CLT9 min
The four notions of convergence for random variables, which implications hold between them, and the two limit theorems everything downstream depends on.
Implications between modes of convergenceWeak law of large numbersStrong law of large numbers
Problems (10)
Open in the browser| Solved | Problem | Type | Difficulty | Time |
|---|---|---|---|---|
| Pairwise independence does not imply independence | Proof | 15 min | ||
| When is the union bound tight? | Proof | 15 min | ||
| The weak law needs only uncorrelatedness | Proof | 15 min | ||
| Conditional expectation minimises mean squared error | Proof | 30 min | ||
| From an exponential tail bound to almost sure convergence | Proof | 30 min | ||
| Law of total variance and its use | Proof | 30 min | ||
| Moments from tail bounds via the layer-cake formula | Proof | 30 min | ||
| Separating the four modes of convergence | Proof | 30 min | ||
| The 1/n variance rate requires uncorrelated samples | Numerical | 30 min | ||
| Why the CLT cannot be used for tail bounds | Numerical | 30 min |