Probability (Measure-Theoretic)

Year 4 · 6 sections · 12 concepts · 11 practice questions

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What's covered

free1 · Probability spaces and random variables2 concepts

Definition 1.1 · Probability space and random variables
Definition 1.2 · Expectation

2 · Independence2 concepts

Definition 2.1 · Independence
Theorem 2.2 · The Borel–Cantelli lemmas

3 · Inequalities and modes of convergence2 concepts

Theorem 3.1 · Markov, Chebyshev and Jensen
Theorem 3.2 · Modes of convergence

4 · Laws of large numbers2 concepts

Theorem 4.1 · Weak law of large numbers
Theorem 4.2 · Strong law of large numbers

5 · Characteristic functions and the central limit theorem2 concepts

Definition 5.1 · Characteristic functions
Theorem 5.2 · Central limit theorem

6 · Conditional expectation and martingales2 concepts

Definition 6.1 · Conditional expectation
Theorem 6.2 · Martingales and optional stopping

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Definition 1.2 · Expectation

Expectation = Lebesgue integral against $\mathbb{P}$ . Discrete sums and density integrals are special cases. Tail formula: $\mathbb{E}X = \int_0^\infty\mathbb{P}(X > t)\,\mathrm{d}t$ for $X\ge0$ .

Proof

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