For Problem Set #4:
  1. #2: Transformation: z=x+y; w=x-y.
  2. #3: Assume x,y are independent rvs.

    For Problem Set #3:
  1. #4: Joint pdf fa,b,c(A,B,C)= fa(A)fb(B)fc(C)= fx(A)fx(B)fx(C)
    since a,b,c are independent and identically distributed.
  2. #6: What's confusing is fx(X) is a function as well as a desired pdf.
    Also, xi and yi are independent and uniformly distributed in [0,1).

    For countable vs. uncountable sets:
  1. Provide a 1-1 correspondence with a set of known cardinality:
  2. {integers}, {rationals}, {lattice points} are all countable;
  3. interval [0,1), {reals}, (power set of countably infinite set) are all uncountable.

    For conditional probability:
  1. Pr[A|B]=Pr[AB]/Pr[B]=Pr[AB]/(Pr[AB]+Pr[A'B]);
  2. Note the form C/(C+D) in this formula;
  3. Remember to divide by Pr[GIVEN event].

    The Cantor set (covered in recitation):
  1. Significance to EECS 501: The Cantor set is uncountable,
  2. since its endpoints are not rational numbers.
  3. Yet Pr[Cantor set]=0 for the wheel of fortune experiment!
  4. Why are the endpoints irrational? Suppose they were rational.
  5. Then their decimal expansions would be "eventually zero" for some finite N.
  6. But the construction of the Cantor set makes it clear this is not the case.