Probability II
Year 2 · 6 sections · 61 concepts · 128 practice questions
What's covered
free1 · Joint distributions11 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Joint densities and marginals
- Proposition 1.2 · Independence and factorisation
- Theorem 1.3 · Transformation of random vectors
- Corollary 1.4 · Density of a sum (convolution)
- Example 1.5 · Polar coordinates for two normals
- Definition 1.6 · Conditional densities
- Definition 1.7 · Conditional expectation as a random variable
- Theorem 1.8 · The tower property
- Proposition 1.9 · Law of total variance
- Check 1.10 · Check: transformation and conditioning problems
2 · Generating functions11 concepts
- Overview 2 · The plan of this section
- Definition 2.1 · Probability generating functions
- Proposition 2.2 · Moments from the PGF
- Proposition 2.3 · PGFs of independent sums
- Theorem 2.4 · Random sums
- Definition 2.5 · Branching processes
- Theorem 2.6 · Extinction probability
- Definition 2.7 · Moment generating functions
- Definition 2.8 · Characteristic functions
- Theorem 2.9 · Uniqueness and continuity theorems
- Check 2.10 · Check: generating-function computations
3 · Multivariate normal10 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Mean vector and covariance matrix
- Proposition 3.2 · Covariance matrices are positive semidefinite
- Proposition 3.3 · Linear transformations of random vectors
- Definition 3.4 · The multivariate normal distribution
- Proposition 3.5 · MGF of the multivariate normal
- Theorem 3.6 · Density of the multivariate normal
- Theorem 3.7 · Uncorrelated jointly normal variables are independent
- Proposition 3.8 · Conditional distributions of the bivariate normal
- Check 3.9 · Check: multivariate normal calculations
4 · Convergence and limit theorems10 concepts
- Overview 4 · The plan of this section
- Definition 4.1 · Modes of convergence
- Theorem 4.2 · Relations between modes of convergence
- Example 4.3 · Counterexamples separating the modes
- Theorem 4.4 · Weak law of large numbers
- Theorem 4.5 · Strong law of large numbers
- Theorem 4.6 · Central limit theorem (proof via MGFs)
- Proposition 4.7 · Slutsky's lemma and the continuous mapping theorem
- Corollary 4.8 · The delta method
- Check 4.9 · Check: proving convergence of random variables
5 · Markov chains11 concepts
- Overview 5 · The plan of this section
- Definition 5.1 · Markov chains and transition matrices
- Proposition 5.2 · n-step transitions (Chapman–Kolmogorov)
- Definition 5.3 · Communicating classes, recurrence and transience
- Proposition 5.4 · Recurrence criteria
- Theorem 5.5 · Hitting probabilities and times by first-step analysis
- Example 5.6 · Gambler's ruin
- Definition 5.7 · Stationary distributions
- Definition 5.8 · Periodicity
- Theorem 5.9 · Convergence to equilibrium
- Check 5.10 · Check: analysing a Markov chain
6 · Poisson process8 concepts
- Overview 6 · The plan of this section
- Proposition 6.1 · The memoryless property
- Definition 6.2 · The Poisson process
- Theorem 6.3 · Inter-arrival times are i.i.d. exponential
- Proposition 6.4 · Arrival times have gamma distributions
- Proposition 6.5 · Given the count, arrivals are uniform
- Theorem 6.6 · Superposition and thinning
- Check 6.7 · Check: Poisson process problems
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Proposition 1.2 · Independence and factorisation
Independent ⇔ joint density factorises (on a product region). Triangle support ⇒ dependent.
Proof
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