Victor Angielczyk's Final Exam Review Site

Focus Areas

Sample space vs event space; random variables; counting; conditional probability.

Competencies

• Define a probability model: sample space \(\Omega\), event space \(\mathcal{E}\), measure \(P\). ?
• Do counting-based probability cleanly. ?
• Translate "word problems" into sets, compute \(P(E), P(E \cap F)\). ?

Focus Areas

Law of Total Conditional Probability; independence; Bayes’ Theorem; Naïve Bayes.

Competencies

• Use independence: \(P(A \cap B) = P(A)P(B)\). ?
• Apply Law of Total Probability. ?
• Use Bayes’ rule (odds update intuition). ?
• Do Naïve Bayes classification. ?

Focus Areas

Continuous RVs; PDFs/CDFs; integration; key distributions; z-scores.

Competencies

• Work with PDFs and CDFs. ?
• Models: Uniform, Exponential, Gaussian. ?
• Compute z-scores. ?

Focus Areas

EV formulas; linearity; variance identity; SD.

Competencies

• Compute expectations (sum / integral). ?
• Use linearity of expectation. ?
• Compute variance/SD. ?
• Know common EV/Var pairs. ?

Focus Areas

Joint/marginal/conditional distributions; covariance/correlation.

Competencies

• Go between joint \(\leftrightarrow\) marginal/conditional. ?
• Test independence via factorization. ?
• Compute covariance \(\text{Cov}(X,Y)\). ?

Focus Areas

LLN, convergence in distribution, CLT approximations.

Competencies

• State/use WLLN. ?
• Distinguish SLLN from WLLN. ?
• Use CLT via standardized sums \(Z_k \to \mathcal{N}(0,1)\). ?

Focus Areas

Transition matrices, stationarity, convergence.

Competencies

• Use transition matrix \(P\). ?
• Compute stationary distribution \(\pi = \pi P\). ?
• Know convergence conditions. ?

Focus Areas

Likelihood, MLE/MAP, least squares, regularization.

Competencies

• Write likelihood \(L(\theta)\); define MLE. ?
• Understand MAP. ?
• Linear regression via least squares. ?
• Ridge regularization. ?

Focus Areas

Multivariate Gaussian; GMM; EM algorithm.

Competencies

• Multivariate Gaussian density \(\mathcal{N}(x \mid \mu, \Sigma)\). ?
• Gaussian mixtures. ?
• EM equations: responsibilities and updates. ?

Focus Areas

Eigenvectors/values, projection, dimensionality reduction.

Competencies

• Center data, form covariance matrix. ?
• Eigen-decomposition: \(\lambda, v\). ?
• Project onto top components. ?

Focus Areas

Max-margin, hard/soft margin, dual form, kernels.

Competencies

• Soft-margin primal; \(C\) tradeoff. ?
• Dual form. ?
• Kernel trick \(K(x,z)\). ?