Plan for 88-262, General Probability, 1998/9

There are 11 lectures. The list below gives the subjects to be presented in each one:
  1. Classical Probability and Combinatorics. Definition of classical probability. Simple problems in combinatorics. Binomial coefficients and properties. Binomial theorem.
  2. Axioms of Probability. Algebra of events (without definition of field of events). Axioms of a probability measure and examples, simple implications of the axioms, inclusion-exclusion theorem and applications in combinatorics.
  3. Conditional probability. Definition, law of total probability, Bayes' theorem. Independence. Partitions, law of total probability and Bayes' theorem for general partition.
  4. Discrete Random Variables. Definition. Computation of probabilities, Expectation (variance deferred till lecture 9), medians and modes. Function of a random variable. (short lecture)
  5. The Classical Discrete Distributions. Binomial, Poisson, geometric and negative binomial.
  6. Continuous Random Variables. Density function, cummulative distribution function, computation of probabilities, expectation, medians and modes, function of a random variable.
  7. The Classical Continuous Distributions (Two lectures) Uniform, exponential, normal and gamma/chi squared. For normal include explanation of how to use to approximate binomial and Poisson.
  8. Expectation. Expectation of function of a random variable (discrete and continuous cases). Variance. Moments. Computation of variance for classical distributions. (short lecture - hopefully start material for next one)
  9. Joint Distribution of Two Variables (discrete case only). Joint distribution, marginal distributions, conditional distributions. Independence of random variables. Distribution of sum of independent random variables. Scalar function of a pair of random variables. Formula for expectation of a scalar function (without proof). Expectation of sum of random variables. E[g(X)h(Y)] when X,Y independent. Variance of sum of independent random variables. (long lecture)
  10. Tchebycheff inequality, weak law of large numbers. Central Limit Theorem: statement of theorem, examples, proof via moment generating functions (assuming mgf characterizes distribution). (short lecture)