MATH313/413 Probability Theory
List of most important denitions and results
• Axioms of sigma-algebras; axioms of probability; probability space
• Conditional probability; Bayes' theorem and its proof
• Mutually independent events
• Denition of a random variable
• Expectation and moments; how to compute expectation of continuous and
discrete random variables and of a function of a random variable
• Distribution and cumulative distribution function; continuous random variables
and probability density function; discrete random variables and probability
mass function
• Normal, Exponential and Uniform distributions: probability density functions,
expectations, variances
• Bernoulli and Poisson distributions: probability mass functions, expectations,
variances
• Denition of independent random variables; summation of normal and
Poisson independent random variables
• Variance; Chebyshev's inequality and its proof
• Convergence in probability; convergence in distribution (weak convergence)
• Weak Law of Large Numbers and its proof
• Characteristic functions; moments and dierentiability; replication of random
variables; Levy's continuity theorem
• Central Limit Theorem and its proof
• Poisson Approximation