Prelim 2 topics

This list of topics is not necessarily complete. Everything covered through Euler's theorem (Monday 10/24) will be in scope. Material covered since the first prelim, and material covered on the homework will be given heavier emphasis.

Some related questions from previous semesters are here, (solutions).

• Probability
  • Basic definitions: sample space, event, probability measure
  • Condiitonal probability: definition
  • Probability trees
  • Independence, mutual independence vs. pairwise independence
  • Bayes' Theorem and law of total probability
  • Random variables: definition, independent random variables
    • Probability distributions: uniform distribution, binomial distribution
    • Expectation: definition, linearity of expectation
    • Markov's inequality, Chebyshev's inequality, variance
• Number theory
  • use euclidean division in definitions and proofs
  • base b representation:
    • definitions, algorithms for base-b arithmetic, know existness and uniqueness
  • apply algorithms contained in inductive proofs
    • e.g. eucidean division, base-b representation, Bezout coefficients
  • determine whether functions defined using equivalence classes are well-defined or not
  • definitions:
    • quotient, remainder, a|b, linear combination, equivalence class, unit
  • theorems:
    • division algorithm, base b representation, euclid's gcd algorithm, Bezout coefficients, Euler's theorem, modular arithmetic operations are well-defined