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280C Homework Assignments (Spring 2007)

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Click here (updated 6/11/2007) to see solutions to old homework assignments.

 

  1.  Homework #1 problems, due Friday, April 13, 2007. See beginning of the Lecture Notes for the list of homework problems.
     
  2.  Homework #2 problems, due Friday, April 20, 2007. See beginning of the Lecture Notes for the list of homework problems. (Warning: some of the exercise numbers have changed in the revised version of the lecturer notes.  Please see the beginning of the new version of the notes to see the current numbering.)
     
  3. Homework #3 problems, due Friday, April 27, Monday, April 30, 2007. See beginning of the Lecture Notes for the list of homework problems. Please also do Exercise 21.2.
     
  4. Homework #4 problems, due Monday, May 7, 2007. See beginning of the Lecture Notes for the list of homework problems.
     
  5. Homework #5 problems, due Monday, May 14, 2007. See beginning of the Lecture Notes for the list of homework problems.
     
  6. Homework #6 problems, due Friday, May 25, 2007. See beginning of the Lecture Notes for the list of homework problems.
     
  7. Last! Homework #7 problems, due Friday, June 8, 2007. See beginning of the Lecture Notes for the list of homework problems.

 

280B Homework Assignments (Winter 2007)

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Click here (updated 2/23/2007 to see solutions to old homework assignments.

Unless otherwise noted, all problems are from Resnick, S. A Probability Path, Birkhauser, 1999.

  1. due Monday, January 22, 2007
    bullet Hand in from p. 114 : 27
    bullet Hand in from p. 196 : 5, 7
    (Hint for 7: X_n has the same distribution as (\sigma_n) N, where N is standard Normal.)
    bulletHand in from p. 234--246: 12, 16, 33, 36, 42
    Comments:
        For 12, let {U_n:n=0,1,2,...} be i.i.d. random variables uniformly distributed on (0,1) and
    take X_0=U_0 and X_(n+1)=X_n*U_(n+1).
        For 36, assume each X_n is integrable! Hints: use the assumptions to bound E[X_n]
    in terms of E[X_n:X_n<=x]. Then use the two series theorem.
     
  2. Homework #2 problems, due Monday, January 29, 2007. See beginning of the Lecture Notes for the list of homework problems.
     
  3. Homework #3 problems, due Monday, February 5, 2007. See beginning of the Lecture Notes for the list of homework problems.
     
  4. Homework #4 problems, due Friday, February 16, 2007. For the list of homework problems, see the beginning of the Lecture Notes: Notes_N11_2p.pdf or Notes_N11_1p.pdf.
     
  5. Homework #5 problems, due Friday, February 23, 2007. For the list of homework problems, see the beginning of the Lecture Notes: Notes_N12_2p.pdf and Notes_N12_1p.pdf.
     
  6. Homework #6 problems, due Friday, March 2, 2007.Monday, March 5, 2007.  For the list of homework problems, see the beginning of the Lecture Notes: Notes_N14_2p.pdf or Notes_N14_1p.pdf .
     
  7. Homework #7 problems, due Monday, March 12, 2007.  For the list of homework problems, see the beginning of the Lecture Notes:Notes_N16_2p.pdf and Notes_N16_1p.pdf. For Resnick problem 10.14, please use the filtration generated by the {Y_n}.
     
  8. Homework #8 problems, due Wednesday, March 21, 2007 by 11:00AM!.  For the list of homework problems, see the beginning of the Lecture Notes:Notes_N16_2p.pdf and Notes_N16_1p.pdf.

 

 

 

280A Homework Assignments (Fall 2006)

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Click here (updated 12/1/2006) to see solutions to old homework assignments.

Unless otherwise noted, all problems are from Resnick, S. A Probability Path, Birkhauser, 1999.

  1. p. 20-27 (due Friday, September 29, 2006)
    bullet Look at: 9, 12, ,19, 27, 30, 36
    bullet Hand in: 5, 17, 18, 23, 40, 41
     
  2. p. 63-70 (due Friday, October 6, 2006)
    bullet Look at: 18
    bullet Hand in: 3, 6, 7, 11, 13 and  the following problem.

    Exercise 2.1 Referring to the setup in Problem 7 on p. 64 of Resnick,
    compute the expected number of different coupons collected after
    buying n boxes of cereal.
     
  3. due Friday, October 13, 2006
    bullet Look at from p. 63-70: 5, 14, 19
    bulletLook at lecture notes: exercise 4.4 and read Section 5.5
    bullet Hand in from p. 63-70:  16
    bulletHand in lecture note exercises:  4.1-4.3, 5.1 and 5.2.
    Homework #3 Questions with Answers
     
  4. due Friday, October 20, 2006
    bullet Look at from p. 85--90:  3, 7, 12, 17, 21
    bullet Hand in from p. 85--90:  4, 6, 8, 9, 15
    Also hand in the following exercise.

     Exercise 4.1 Suppose {fn }is a sequence of Random Variables on
    some measurable space.  Let B be the set of     w such that fn (w) is convergent
    as n tends to infinity. Show the set B is measurable,
    i.e. B is in the sigma -- algebra.
     
  5. due Friday, October 27, 2006
    bullet Look at from p. 110--116:  3, 5
    bullet Hand in from p. 110--116:  1, 6, 8, 18, 19
     
  6. due Friday, November 3, 2006
    bullet Look at from p. 110--116:  3, 5, 28, 29
    bullet Look at from p. 155--166:  6, 34
     
    bullet Hand in from p. 110--116:  9, 11, 15, 25
    bullet Hand in from p. 155--166:  7
    bulletHand in lecture note exercise:  7.1.
     
  7. due Friday, November 10, 2006  Monday, November 13, 2006
     
    bullet Look at from p. 155--166:  13, 16, 37
     
    bullet Hand in from p. 155--166:  11,  21, 26
    bulletHand in lecture note exercises:  8.1, 8.2, 8.19, 8.20
    Please see the comments and corrections for this assignment.
     
  8. due Monday, November 20, 2006  Wednesday, November 22, 2006
    Monday, November 27, 2006
     
    bullet Look at from p. 155--166:  19, 34, 38
    bullet Look at from p. 195--201:  19, 24
     
    bullet Hand in from p. 155--166:  14,  18 (Hint: see picture given in class.), 22a-b
    bulletHand in from p. 195--201:  1a,b,d,  12, 13, 33 and 18 (Also assume EX_n = 0)*
    bulletHand in lecture note exercises:  9.1

    * For Problem 18, please add the missing assumption that the random variables should have mean zero. (The assertion to prove is false without this assumption.)  With this assumption, Var(X)=E[X^2].  Also note that  Cov(X,Y)=0 is equivalent to E[XY]=EX*EY.
     

  9. due Noon, on Wednesday, 12/6/2006
     
    bullet Look at from p. 195--201:  3, 4, 14, 16, 17, 27, 30
     
    bulletHand in from p. 195--201:  15 (Hint: |a-b|=2(a-b)+-(a-b). )
    bulletHand in from p. 234--246: 1, 2 (Hint: it is just as easy to prove a.s. convergence), 15