Topics on Final Exam

The exam will be closed book and closed notes. No electronic devices are allowed. The exam will cover only subjects that were discussed in lecture. The following will NOT be on the exam.

• Minutiae of MATLAB. The MATLAB plotting functions.
• Gram-Schmidt orthogonalization algorithm
• Algorithm for inverting a matrix.
• Determinants for matrices larger than 2x2.
• Any questions that require any use of calculus.
• Frequentist definitions of confidence intervals or hypothesis testing.
• Any new subjects introduced in the last lecture 8/1.

You will not be asked to write proofs.

The exam will have two parts; a short answer part of questions you should be able to answer quickly, and a number of small (two or three line) MATLAB programming problems. I will not deduct points for trivial syntax errors in the MATLAB programs, but otherwise I expect them to be correct.

List of topics

• MATLAB: Basic functionalities. (Notes, Chap. 1)
• Vectors. (Chap 2, except section 2.6 on plotting)
  • Basic operations
  • Dot product. Orthogonality. Cosine formula.
  • Basic applications.
• Matrices (Chap 3)
  • Basic operations
  • Multiply matrix times vector
  • Matrix multiplication
  • Linear transformations
• Vector spaces (Chap. 4 section 4.1)
  • Vector spaces
  • Linear independence
  • Basis. Dimension
  • Vector space sum. Complement. Orthogonal complement.
  • Linear transformations: Rank, image, null space, inverse.
  • Categories of systems of linear equations.
• Solving systems of linear equations. (Chap 5, sections 5.1, 5.2, 5.4)
  • Row-echelon form
  • Gaussian elimination
  • MATLAB. You should know the built-in functions M\C, rank(M), null(M). You don't need to know the 6 different things M\C does.
• Geometry (Chap 6, except 6.4.7 and 6.4.8).
  • Points, lines, and planes
    • 3 representational systems for lines and planes
    • Problems: identity, conversion, incidence, generation, parallelism, and intersection.
    • Projection of a point onto a line or plane. Distance from a point to a line or plane.
  • Geometric transformations
    • Natural and homogeneous coordinates
    • Translation
    • Rotation
    • Rigid mappings
    • Reflections
    • Affine transformations
• Change of basis (Chap 7, sec 7.1-7.3)
  • Change of coordinate system
• Probability (Chap. 8)
  • Sample spaces
  • Axioms
  • Bayes' Law
  • Independence and conditional independence
• Numerical Random Variables (Chap 9)
  • Random Variables
  • Joint distribution. Marginal distribution
  • Expectation, variance, and standard deviation
  • Basic theorems including the Central Limit Theorem. I won't ask you to parrot these back or to prove them, but I may ask questions that require that you use them.
  • Decision theory: Maximize expected utility.
  • Specific distributions.
    • Bernoulli. Know the distribution, mean, standard deviation, significance.
    • Binomial. Know the distribution, mean, standard deviation, significance.
    • Uniform continuous. Know the density, the mean, the significance. You do not need to know the standard deviation.
    • Gaussian. Know the density, the mean, the standard deviation, the significance.
• Confidence Intervals (Chap 11, section 11.1). Bayesian definition (11.3)
• Monte Carlo methods (Chap 12.1-12.3, 12.6)
  • Finding Area
  • Counting
  • Probabilistic problems