MA395-A Final Exam Study Guide

The final will be comprehensive, including:

• The following material from the text:
  • Chapter 2
  • Chapter 3 (excluding Section 10)
  • Chapter 4 (excluding Section 11)
  • Chapter 5
  • Chapter 6
  • Chapter 7 (sections 2,3 and 4 only)

• The following supplemental material (posted in the notes and handouts section)
  • The multivariate normal density function
  • The variance-covariance matrix
  • Expectation, variance, and covariance of linear combinations of random variables
  • Quadratic forms and their relation to the chi-square distribution
  • The proof of the central limit theorem (also posted under notes and handouts)

• Section 1: Proofs. This section will contain:
  • A proof you have seen before - one of the following:
    • Markov's theorem for a discrete random variable (see the proof given in class)
    • The central limit theorem (a proof with more detail than required is posted in the notes and handouts section)
  • Two additional short proofs which you may or may not have seen before.

• Section 2: Applications (12 problems choose 10) These will be similar to the problems on the assignments. The best way to prepare for this part of the exam is to review the solutions to the assignments (especially the more recent ones). Time constraints dictate that the problems on the exam have to be less difficult (on average) than those in the assignments, so if you know how to do the assigned problems, you should be well prepared.

• Section 3: A (humongous) matching section with terms and definitions.

  You should be familiar enough with the definitions of the following terms to recognize them in a matching problem:

  • the joint density function of two random variables X and Y
  • the variance of a random variable
  • conditional density of Y given X=x
  • the k^th moment of a random variable Y about the origin
  • E(Y²)
  • the variance of t'W where t is a vector and W is a vector of random variables
  • the probability density function of min(Y₁,...,Y_n), where the Y_i are IID
  • a bilinear form (an expression of the form x'Ay)
  • marginal density
  • rank of a matrix
  • k^th central moment of a random variable Y
  • idempotent matrix
  • the covariance of two random variables X and Y
  • the moment-generating function m(t) of a random variable
  • the expected value of a linear combination of random variables t'W
  • the moment-generating function of aY+b where Y is a random variable with moment-generating function m(t)
  • the expected value of a random variable
  • independent random variables
  • the expected value of A'W where A is a matrix and W is a random vector
  • the variance of A'W where A is a matrix and W is a random vector
  • quadratic form (an expression of the form x'Ax)
  • the cumulative distribution function of a random variable
  • an event
  • a sample space
  • the Kolmogorov axioms
  • a discrete sample space
  • a compound event
  • a simple event
  • an experiment
  • the Borel sigma-algebra on R
  • a probability triple
  • a probability measure
  • a measure space
  • a measure
  • a measurable space
  • the power set P(S)
  • DeMorgan's law
  • equivalent sets
  • a proper subset
  • the Cartesian product
  • the inverse image of a set under a function f^-1[T]
  • Cantor's definition of an infinite set
  • an uncountable set
  • a sigma-algebra on a set S
  • an algebra of sets on a set S
  • the sigma-algebra generated by a subset T of the power set of a set S
  • Markov's inequality
  • Chebychev's inequality
  • the t distribution (Def. 7.2)
  • the F distribution (Def. 7.3)

All formulas listed inside the back cover of the text will be supplied, in addition to the equivalent expressions for the multivariate normal distribution (density function, moment-generating function). Any formulas needed from the supplementary material will also be supplied.