Exam 1: Sets and Combinations

  Ch 1: sets
    1.1: set operations:
      intersection, union, complement, cartesian product.
    1.2: Venn diagrams
      + counting
        * partitions (disjoint unions): n(A "+" B) = n(A) + n(B),
            where "+" means disjoint union.
        * cartesian products: n(A x B) = n(A)*n(B)
      + deMorgan's laws:
        * (A union B)' = A' intersect B'
        * (A intersect B)' = A' union B'
    1.3: counting
      * n(A union B) = n(A) + n(B) - n(A intersect B).
    1.4: trees: counting possible outcomes of a sequence of experiments
        by multiplying number of possible outcomes
        of a sequence of experiments.

  Ch 2: Combinatorics (counting)
    2.1: Probabilities
      * sum of probabilities of all outcomes is 1.
      * Pr(E) = n(E)/n(sample space) if all outcomes are equally likely
    2.2: Permutations (order matters)
      P(n,r) = n!/(n-r)! = n*(n-1)*...*(n-r+1)
        = permutations of n objects taken r at a time
    2.3: Combinations (order doesn't matter)
      C(n,r) = P(n,r)/r!
        = combinations of n objects taken r at a time

Exam 2: Probability

  Ch 3: Probability of events
    3.1: axioms and properties of probability
      * probability measures the size of a set of outcomes.
      * axioms:
        1. 0 <= Pr[E] <= 1
        2. Pr[S] = 1
        3. Pr[E "+" F] = Pr[E] + Pr[F].
      * properties:
        1. Pr[E'] = 1 - Pr[E]
        2. Pr[E union F] = Pr[E] + Pr[F] - Pr[E intersect F]
     3.2: conditional probability
       * Pr[A|B] := Pr[A and B]/Pr[B]
       * A and B are independent if Pr[A and B] = Pr[A]*Pr[B],
           i.e., knowledge of one event gives no information
           about the probability of the other.
     3.3: trees (to depict outcomes of stochastic processes)
     3.4: Bayes probabilities
       * the probability of a leaf of a tree is the
         product of the conditional probabilities along
         the branches to that leaf;
       * Bayes' formula expresses this idea:
           Pr[A|B] = Pr[B|A]Pr[A]/Pr[B] (for two events).
       * Bernoulli process: for n independent Bernoulli trials
           each with probability of success p and probability
           of failure q = 1-p,
         Pr[r successes] = C(n,r) p^r q^(n-r)

  Ch 4: Random Variables (random numbers)
    4.1: probability density function
      * binomial random variable = number of successes of Bernoulli process
    4.2: Expected value
      * definition
        Let X = random variable with k outcomes.
        Let x_j = outcome number j.
        Let p_j = P(X=x_j).
        Then E[X] = x_1*p_1 + x_2*p_2 + ... + x_k*p_k.
      * expectation of binomial random variable:
          E[X] = n*p, (n trials each with success probability p)
      * variation and standard deviation:
          Let m = E[X].
          Var[X] = (x_1 - m)^2*p_1 + (x_2 - m)^2*p_2 +...+ (x_n - m)^2*p_n.
          standard deviation = sigma = sqrt(Var[X]).
      * variance of binomial random variable:
          Var[X] = n*p*q.
      
Exam 3: Linear Algebra

  Ch 5: systems of equations
    5.1: lines
      * slope = m = rise over run = (y_2 - y_1)/(x_2 - x_1)
      * use point-slope formula and/or slope-intercept formula
          to find the equation of a line through two points
    5.2: linear systems
      * substitution method
      * reduction method (combining equations)
    5.3 linear systems in many variables (*important*)
      * 3D graphs of planes
      * matrix representation
      * row operations
      * row reduction
      * reduced form
      * solution sets
        + unique solution
        + overdetermined case (inconsistent system)
        + underdetermined case (infinite family of solutions)

  Ch 6: Matrix algebra
    - matrix addition and multiplication
    - matrix identity
    - matrix inverse
    - using matrix inverse to solve linear systems

Exam 4: Applications

  Ch 7: Linear programming
    - feasible sets
    - evaluation at corner points and auxiliary points

  Ch 8: Markov Chains
    - state transition matrix
    - state vectors
    - regular markov chains
    - stable state vector

Final material

  Ch 9: financial math
    - compound interest
    - present and (future) amount
      - present value of annuity
      - amount of annuity
      - payment of ammortized loan
      - payment of sinking fund