MCS 341 EXAM 4
  • When: Available November 22. Due November 24
  • What: An examination consisting of problems related to Chapter 5, Multivariate Probability Distributions and supplementary class notes.
  • How: Take-home and closed-book, but with one new 3"-by-5" note card allowed. (You may also use your note cards from the previous exams.)
    TOPICS
  • Bivariate and multivariate probability distributions
    • The joint distribution function for jointly distributed random variables Y1, ..., Yn, i.e., r.v.s defined on the same probability space, F(y1, ..., yn) = P(Y1<y1, ..., Yn<yn), is always defined.
    • Jointly distributed "continuous" random variables are those whose joint distribution function is continuous.
    • Some, but not all, jointly distributed continuous r.v.s have a joint probability density function. Their joint distribution function and other probabilities may be calculated by integrating the joint pdf.
    • Jointly distributed discrete random variables have a joint probability function.
  • Marginal and conditional probability distributions
    • The marginal distributions are found by summation (discrete case) or integration (pdf case) of the joint probability function or joint pdf.
    • The conditional probability function or pdf is found by dividing the joint probability function or pdf by the appropriate marginal function.
  • Independent random variables
    • Definition
    • Tests for independence, including Theorem 5.5
    • Note the clarifications given in class.
  • The expected value of a function of random variables: the Law of the Unconscious Statistician
  • Special theorems, especially linearity
  • The covariance of two random variables
    • Definition and computing formulas
    • Correlation coefficient
    • Properties of covariance and correlation.
    • Relationship between uncorrelatedness and independence.
  • Expected value, variance, and covariance of linear combinations of random variables
  • The multinomial probability distribution
    • Joint probability function
    • Expected values, variances, and covariances
  • The bivariate and multivariate normal distributions will not be covered on this exam.
  • Conditional expectations