Plan for 88-262, General Probability, 1998/9
There are 11 lectures. The list below gives the subjects to
be presented in each one:
- Classical Probability and Combinatorics. Definition of classical
probability. Simple problems in combinatorics. Binomial coefficients
and properties. Binomial theorem.
- 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.
- Conditional probability. Definition, law of total probability,
Bayes' theorem. Independence. Partitions, law of total probability
and Bayes' theorem for general partition.
- Discrete Random Variables. Definition. Computation of probabilities,
Expectation (variance
deferred till lecture 9), medians and modes. Function of a random
variable. (short lecture)
- The Classical Discrete Distributions. Binomial, Poisson, geometric
and negative binomial.
- Continuous Random Variables. Density function, cummulative distribution
function, computation of probabilities, expectation, medians and modes,
function of a random variable.
- 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.
- 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)
- 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)
- 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)