MCS 256 Final Exam

The final examination is scheduled for 1:00-3:00 p.m. Monday, May 21, in our classroom.

Your may use one new 4"-by-6" handwritten note card or two 3"-by-5" handwritten note cards plus the note cards you prepared for the previous exams. I'll provide a table of discrete distributions and Rosen's table, "Useful Generating Functions."

This exam will be comprehensive with special emphasis on material covered in the last three weeks.

MCS 256 Course Outline

• Introduction to discrete math
  including Mathematical induction
• Difference calculus
• Summation calculus
• Asymptotics
• Combinatorics: how to count
  and elementary probability
• Probability (continued)
  and probability generating functions
• Generating functions
• Recurrence relations

Recent topics: Recurrence relations and generating functions

• Ordinary generating functions (ogfs)
  • Definition and
    1-1 correspondence between ogfs and sequences
    • If G(x) = g₀ + g₁x + g₂x² + ..., then g_n = [xⁿ]G(x) = (coefficient of xⁿ in the series expansion of G(x)) = G⁽ⁿ⁾(0)/n!.
    • Standard problem #1: Given a sequence, find its generating function.
    • Standard problem #2: Given a generating function, find the sequence that generates it.
  • Basic ogfs
    • Geometric series
    • Binomial series
    • Exponential series
    • Table of useful generating functions
  • Probability generating functions are ogfs of sequences p₀, p₁, p₂, ... of probabilities.
    Review properties of, and formulas for, pgfs.
  • Basic manipulations
    • Adding, subtracting,multiplying ogfs
    • Shifting sequences
    • Multiplying terms by a power or factorial power of the index n is related to differentiation and multiplication of ogfs by x.
• Recurrence relations and generating functions
  • Standard problem: Given a recurrence for sequence g_n, find an equation for the generating function G(x) of the sequence. Then you may be able to find a formula for g_n.
• Linear recurrences with constant coefficients
  • Such recurrences are equivalent to linear difference equations with constant coefficients, and vice versa.
  • Solution of homogeneous linear recurrences via the characteristic equation: The procedure was outlined in the course.
  • Solution of nonhomogeneous recurrences: Recipes were given in class.
• Divide-and-conquer algorithms and recurrence relations
  • Solutions (in special cases at least)
    • By change of variable, e.g., u_k = f(b^k), to get a fixed-order recurrence relation
    • By backtracking
  • The "Master Theorem"

  Here's an old MCS 256 exam and my solutions.
  Here's an old MCS 256 exam on generating functions and recurrence relations and the answers. Note that problem 3(b) involves exponential generating functions, which will not be on your exam.
  More review problems
  and solutions