Mathematics and Statistics

Information

Part I: Mathematics

Lecture 1: Differentiation

• constant factor rule; sum rule; product rule; chain rule; derivatives of special functions, higher order derivatives; Taylor’s theorem.

Lecture 2: Integration

• integral as signed area of the region bounded by its graph; Riemann integral; improper integrals; change of variables; multiple integrals; special integrals.

Lecture 3: Matrix algebra

• vectors and matrices; rank; positive (semi) definite matrices; eigen value decomposition; singular value decomposition; trace; determinant.

Part II: Optimization

Lecture 4: Quadratic programming

• univariate quadratic programming; univariate  optimization; multivariate optimization; Newton method; quasi-Newton methods

Part III: Probability theory

Lecture 5: Random events and random variables

• axioms of probability; independent events; law of total probability, Bayes’ law; random variables; cumulative distribution function; quantiles and percentiles; density functions; independent random variables.

Lecture 6: Expectation

• expectations and variances;functions of random variables; linear functions of random variables; random samples.

Lecture 7: Common distributions

• binomial distribution; Poisson distribution; uniform distribution; exponential distribution; normal distribution; normal probability plots; lognormal distribution; chi-squared distribution; Student’s t-distribution. F-distribution; heavy-tailed distributions.

Part IV: Statistics

Lecture 8: Inference

• point estimation; interval estimation; hypothesis testing; P-values; confidence intervals.

Lecture 9: Linear regression I

• linear regression model; least squares; the hat matrix; error sum of squares.

Lecture 10: Linear regression II

• normality assumption; partial t-test; general F-test (pivotal quantity, test statistic; confidence area).