HOMEWORK:
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HW) (click
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- Week 1: 9/8 - 9/10:
- Read Chapter 1: Sections 1.1, 1.2,
1.3, 1.4, 1.5.
- My notes for chapter 1
are online here, and the slides from
the first lecture (9/10/2009) are here.
- Homework: Due Thursday, 9/17 (but
as this is the first assignment, no late penalty if you put it in my mailbox
by 10am on Friday the 18th):
Section 1.3: #2, #3, #5; Combinatorics: (1) There are 2n people who enter as
n pairs of two. The people
are then randomly matched in pairs. What is the probability everyone is matched
with their initial partner? There are two ways to interpret this problem;
either is fine so long as you state which interpretation. In one
interpretation, say there are n people from Williams and n from Amherst,
matched in n pairs with each pair having someone from Williams and someone
from Amherst. In the new matching, you must match someone from Williams with
someone from Amherst. In the other interpretation, anyone can be matched
with anyone. You may solve either problem, just clearly state which one you
are doing (not surprisingly, the answers differ). (2) Consider n people ordered 1, 2, ..., n. We randomly
assign another ordering to these people -- what is the probability at least
one person is assigned the same number twice? Section 1.4: #2, #4. Section
1.8: #2, #4, #6, #12.
- Suggested Problems: Section 1.3:
#1, #6; Section 1.4: #5, #6.
- Week 2: 9/15 - 9/17:
- Read Chapter 1: Sections 1.5, 1.7.
- My notes for chapter
1
are online here, my notes
for chapter 2 are here.
-
Beamer slides of lecture 2 (9/15/2009) are available here
Beamer slides of lecture 3 (9/17/2009) are
available here.
-
Do quiz 1 (you do not need to
write it up). The solutions immediately follow the questions. No quiz
for Thursday's class as HW is due.
- Homework: Due Thursday 9/24 (though
you may place in my mailbox anytime up till 10am on Friday 9/25):
Section 1.5: #1, #2, #4 (also determine if it is true if p is not prime),
#8. Section 1.7: #1, #3 (hint: you can solve this without using
difference equations!), #4. Section 1.8: #28 (also determine if we
must have 10% colored, or if we can do more, and generalize to
4-dimensions if possible). Section 2.1: #2, #4, #5c. Section 2.3: #3 (very
important problem for simulating random variables), #4, #5.
- Suggested Problems: Section 1.5:
#3, #5, #6. Section 1.7: #2, #5. Section 1.8: #1, #3, #7, #11, #16, #18,
#20, #23, #24, #29, #31, #32, #35, are #37 and #38 consistent, #39. Section
2.1: #1, #3, #5abd. Section 2.3: #1, #2.
- Week 3: 9/22 - 9/24:
- Read Chapter 1: Section 1.7 (Example
4). Chapter 2: Sections 2.1, 2.3, 2.5. Note we will skip 2.2 and 2.4 for now.
-
my notes for chapter 2 are here.
- Beamer
slides of lecture 4 (9/22/09) are available here.
My notes for lecture 5 (4/24/09) are available
here
-
Do quiz 2 (you do not need to
write it up). The solutions immediately follow the questions. No quiz
for Thursday's class as HW is due.
- Homework: Due Thursday October 1
(though you may place in my mailbox anytime up till 10am on Friday 10/2):
Section 2.5: #2, #6. Section 2.7: #1, #4af, #7, #11, #18. Create two
homework problems and TeX them up. They may be on anything related to
probability; the first one you must be able to solve (and include the solution
in your write-up); for the second, it's fine not to be able to do it (feel
free to include a problem whose solution you'd like to know). I will share the
problems and solutions with the class. Click
here for LaTeX / Mathematica templates;
or click here for a template specific for
writing homework problems.
- Suggested Problems: Section 2.5: #1,
#4. Section 2.7: #2, #3, #8, #12, #13, #15, #20.
- Week 4: 9/29 - 10/1:
- Read: Chapter 3: Sections 3.1, 3.2,
3.3 and Chapter 4: 4.1, 4.2, 4.3.
-
my notes for chapters 3 and 4 are here.
- Beamer
slides of lecture 6 (9/29/09) are available here.
- Homework: Due Thursday October 8
(though you may place in my mailbox anytime up till 10am on Friday 10/9):
Section 3.1: #1ac (hint: famous sum), #3 and Section 4.1: #1b. Section
3.2: # 1, #4 and Section 4.2: #1 (also do when F is uniform on [0,1] and K =
.9); obviously your solution will depend on the unknown distribution F.
Section 3.3: #1, #2, #7 and Section 4.3: #1a, #2.
- Suggested Problems: Section 3.1: #5
and Section 4.1: #1a, #4. Section 3.2: #2, #5 and Section 4.2: #2. Section
3.3: #3, #4, #8 and Section 4.3: #1b, #4, #5.
- Week 5: 10/6 - 10/8:
TAKE HOME EXAM THE FOLLOWING WEEK:
click here for a review sheet
- Week 6: 10/13 - 10/15:
NO CLASS TUESDAY.
- Read: Chapter 3: Sections 3.3, 3.4,
3.5, 3.6 and Chapter 4: Sections 4.3, 4.4, 4.5.
-
my notes for chapters 3 and 4 are here.
- Beamer
slides of lecture 10 (10/15/09); here is the
Mathematica program for Monte Carlo
integration of the n-dimensional sphere's volume.
- Homework: Due Thursday October
22 (though you may place in my mailbox anytime up till 10am on Friday 10/23):
Section 3.5: #2. Section 4.4: #5. Section 3.6: #2, #7. Also TeX up two problems; you must include an answer for the first. Please see
the TeX code from the first problem set (TeX
code is here, compiled PDF is
here). EXTRA CREDIT: Section 4.5: #6.
- Suggested Problems: Section 3.5: #1,
#4. Section 4.4: #2, #3, #4. Section 3.6: #1, #5, #6. Section 4.5: #2.
- Week 7: 10/20 - 10/22:
- Read: Chapter 3: Sections 3.6 and Chapter 4: Sections 4.5,
4.7, 4.8.
-
my notes for chapters 3 and 4 are here.
- Beamer
slides of lecture 11 (10/20/09); slides of
lecture 12 (10/22/09).
-
here is the Mathematica program for generalized prize problem;
here is more info on the
three hats problem.
- Homework: Due Thursday October 29 (though you may place in my mailbox anytime up till 10am on Friday 10/30):
Section 4.7: #2. Section 3.11: #14. Section 4.14: #35, #45bc.
- Suggested Problems: Section 3.11:
#1a, #2, #4, #7, #10, #11, #17, #18, #21, #22, #25, #30. Section 4.14: #18,
#21, #27, #28, #33, #44. Section 4.14: #16.
- Week 8: 10/27 - 10/29
- Read: Chapter 3: Sections 3.8 and Chapter 4: Sections 4.7,
4.8, 4.10.
-
Start reading the notes on generating functions.
-
my notes for chapters 3 and 4 are here.
- Beamer slides of lecture 13
(10/27/09) and
slides of lecture 14 (10/29/09)
- We will do
Pepys's problem,
Section 3.8: #5, in class (unless there is another problem people want to see);
here is the
Mathematica notebook for Pepys'.
- Homework: Due Thursday November 5
(though you may place in my mailbox anytime up till 10am on Friday 11/6):
Section 5.1: #1c, and also find the mean and the variance (formulas for these
are given on pages 151 and 152). Additional problem (1): Let X1,... Xn be
independent random variables having the standard exponential distribution.
Using convolutions, find the density for X1 + X2; more
generally, find the density for X1 + ... + Xn.
Additional problem (2): Find the Fourier transforms of the densities for the
uniform distribution on [0,1] and for the uniform distribution on [-1/2, 1/2].
Additional problem (3). Let X1, ..., Xn be n independent standard normals.
Using convolutions, show that X1^2 + X2^2 is a chi-square distribution with 2
degrees of freedom, and more generally that X1^2 + ... + Xn^2 is a chi-square
distribution with n degrees of freedom. (The chi-square distribution is
defined on page 97.) Finally,
do one and only one of the following:
- create and solve one problem based on
the material we have covered in the past two weeks; if you wish, you may
include a problem for me to solve as well. Unlike previous weeks, this time
you must have a classmate check your problem and solution, and sign off on the
problem. You may of course discuss the problem with several people if you
wish, but everyone must have their problem checked by a classmate; if you
cannot find anyone, let me know. As a reader, your job is to make sure the
problem is clear and the solution is correct and easy to follow. Please list
on your file who looked at your problem (if you want, the two of you may
submit one TeX file).
- Do any three of the suggested
problems for this week.
- Suggested Problems: Section 4.7: #8,
#11. Section 3.8: #3, #4, #6. Section 4.8: #1, #8. Section 5.1: #1a. Section
5.3: #9.
- Week 9: 11/3 - 11/5
- Week 10: 11/10 - 11/12
- Week 11: 11/17 - 11/19: Note final midterm will be a 24 hour take-home
to be done b/w Thursday, November 19 and Sunday, November 29.
- Week 12: 11/24 - 11/26: Note final midterm will be a 24 hour take-home
to be done b/w Thursday, November 19 and Sunday, November 29.
- Read: Benford's law: (you don't need to read all of this; I just want to
assemble all the information in one place)
- Read: MSTD (more sums than differences) Sets: (you don't need to read all
of this; I just want to assemble all the information in one place)
- Beamer
slides of lecture 21 (11/24/09)
- Homework:
HW due Thursday, December 3rd. (though you may submit by 10am Friday the
4th without penalty)
- Suggested Problems: Review suggested problems from throughout the
semester.
- Week 13: 12/1 - 12/3:
Week 14 12/8 - 12/10:
Homework:
HW due Thursday, December 10 (though you may
submit by 10am Friday the 11th without penalty)
Suggested Problems: Review suggested problems from throughout the
semester.