# 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