MCS 341 EXAM 3
  • When: Friday, November 5, 10:30-11:20
  • What: An examination consisting of problems related to Chapter 4, Continuous Random Variables and Their Probability Distributions and supplementary class notes.
  • Where: Our classroom, Olin 219
  • How: Closed-book, but with one new 3"-by-5" note card allowed. (You may also use your note cards from the previous exams.)
    TOPICS
  • Random variables
    • Every random variable Y has a (cumulative) distribution function (cdf): F(y) = P(Y < y).
        Characterizing properties of cdf's: p. 153, including footnote
    • A random variable may be discrete, continuous, or neither.
    • A discrete random variable has a countable range. Every discrete r.v. has a probability function.
    • A random variable Y is continuous iff its cdf is continuous iff P(Y=y) = 0 for all y.
    • Some continuous random variables (but not all) have probability density functions (pdf's). A random variable Y has a pdf iff there exists a nonnegative function f such that for every interval [a, b], P(a < Y < b) = the integral of f over the interval [a, b] iff its cdf F(y) = the integral of f from -infinity to y.
      • By the Fundamental Theorem of Calculus, F'(y) = f(y) where f is continuous.
      • Characterizing properties of a pdf: Nonnegativity & integral over real line is equal to 1.
      • pdf of Y = mX+b [class notes]
  • Expected values (means, expectations)
    • Definition of E(Y) if Y is continuous and has a pdf.
    • Law of the Unconscious Statistician
    • Definition and formulas for the variance
    • Standard deviation
    • Properties of expected values, including linearity
  • Moments and moment-generating functions
    • The kth moment
    • The kth central moment
    • The moment generating function m(t) = E(etY).
      • Calculation via the integral of etyf(y) dy
      • Maclaurin series
      • E(Yk) = m(k)(0).
  • Tchebysheff's Theorem
  • Special continuous distributions
    • The uniform probability distributions
    • The normal probability distributions
    • The gamma probability distributions
      • The exponential probability distributions
      • The chi-square probability distributions
    • The beta probability distributions
    TYPICAL PROBLEMS
  • Check whether a given function is a cdf.
  • Check whether a given function is a pdf.
  • Given a function g(y), find a constant k such that f(y) = kg(y) is a pdf.
  • Given a cdf, find the pdf.
  • Given a pdf, find the cdf.
  • Calculate the probability of Y being in an interval given
    • its cdf;
    • its pdf.
  • Evaluate definite integrals by relating them to integrals of pdf's. See p. 190, 2nd paragraph.
  • Calculate or know the expected value of Y or of g(Y).
  • Calculate or know the variance or standard deviation of Y.
  • Calculate other moments of Y.
  • Calculate or know the moment generating function of Y.