Probability
Undergraduate · Statistics
Syllabus focus
Standard syllabus · Theoretical / proof-based
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Topics typically covered
Standard syllabus
Probability foundations
- Sample spaces, events, and axioms of probability
- Combinatorial counting: permutations and combinations
- Conditional probability and independence
- Bayes' theorem and law of total probability
Random variables and distributions
- Discrete and continuous random variables
- PMFs, PDFs, and CDFs
- Expected value, variance, and moment generating functions (intro)
- Joint, marginal, and conditional distributions
- Covariance and correlation
- Common discrete distributions: Bernoulli, binomial, geometric, Poisson
- Common continuous distributions: uniform, exponential, normal
Limit theorems
- Law of large numbers (statement)
- Central limit theorem and normal approximations
- Chebyshev's inequality (optional)
- Applications to sampling and estimation preview
Theoretical / proof-based
Rigorous probability
- Formal probability spaces and sigma-algebras (introduction)
- Proofs of basic probability laws
- Conditional probability as a measure-theoretic concept (intro)
- Independence: definitions and counterexamples
- Borel–Cantelli lemmas (optional)
Distribution theory
- Transformations of random variables
- Convolutions and sums of independent variables
- Order statistics (introduction)
- Multivariate normal distribution: properties and proofs
- Convergence in probability and in distribution
- Proof sketches of weak law and CLT
Notes
Topics reflect common undergraduate probability syllabi. Theoretical sections mirror math-department probability; standard sections target statistics and applied majors.