Homework
Click the links below to see the assignments.
(Also given below are links to some extra problems and solutions. These are just for you to study, and are not to be turned in for grading.)
- HW 1 (due at class time on Feb. 10,
but grace period until 11:30 PM on 2/10)
- solutions for HW 1
- The 5 parts graded: 1(c), 1(d), 2, 3(b), 5.
- Here is a summary of the scores (for the 7 students who submitted solutions):
- 2 students got scores in the interval [19.5, 19.8],
- 3 students got scores in the interval [16, 18],
- 2 students got scores in the interval [13.4, 14.1].
The median score is 16.8 (out of 20, so 84%).
- Here are some comments about the grading.
- A lot of you lost points (or at least a fraction of a point) for not giving adequate justification for your final answers. With something simple, like part 1(d),
not a lot of explanation is needed,
and for Problem 6, not a lot of words are needed, but you should show a few steps before giving your final answer.
But for more complex problems like 1(c), 2, and 3(b), a somewhat detailed explanation should be
provided for the probabilities,
especially the numerator counts.
(A lot of students could have earned more partial credit if they had properly defined the events they used and better explained their method of solution.)
- Some used a "brute-force" method to obtain a numerator count (in some cases listing every possible outcome which corresponds to an event).
Not only does such an approach lack elegance, but it doesn't "scale up" nicely if one were given larger starting values.
- Be careful to use correct terminology: some students wrote combinations when they would have wrote permutations, and others did the exact opposite.
- In the mathematical portion of your solutions, P should be reserved for the probability function. Don't use P for an event, and don't
write something like P = 2/5. (It's okay to have P(E) = 2/5, given that E has been properly defined, since here P is used as the
probability function, but in P = 2/5, P is being used as a specific probability, and not as the probability function.) Similarly, reserve the use of
S to denote the sample space (don't use it for any other event) ... and keep in mind that S is a set (of possible outcomes) and is not
a number!
- Some students wrote expressions like
P( A ∪ B )C
instead of
P( (A ∪ B )C ).
(One set of parentheses is needed for the event (the complement of A union B), and another set of parentheses is needed for the probability
function.) For this assignment, I only gave a very small deduction for this use of bad notation, but the penalty may be a bit more substantial if such mistakes are
made on an exam.
- A lot of students wrote things like P(A ∪ B) and P(E) without ever defining A, B, or E. I didn't count off much (this time) for such minor abuses of notation, but I may be
more picky in the future.
- Finally, please be sure to drawn boxes around, or highlight, your final answers; use paper which is close to being 8.5 inches by 11 inches,
and staple your sheets together in the upper-left corner.
- Extra Ch. 1 Problems (not to be graded)
- Extra Ch. 2 Problems (not to be graded)
- HW 2 (due at class time on Feb. 17,
but grace period until 11:30 PM on 2/17)
- For the first two problems of HW 2 (as well as perhaps Problem 4 of HW 1), you should generally define events and use some of the results from Ch. 2 (along with some results from Ch. 1), instead of trying to
work them similar to how you worked the 1st 3 problems of HW 1 (mostly based on Ch. 1, and for which each desired probability can be obtained using a ratio of two
counts).
(See the last "bullet" under Presentation of Homework Solutions below for some examples of clearly defining events.)
For the rest of the HW 2 problems, I suggest incorporating some Ch. 3 results
(in addition to using Ch. 1 and Ch. 2 results).
- solutions for HW 2
- The 5 parts graded: 1, 2, 3, 4, 5.
- Here is a summary of the scores:
- 1 students got a perfect score of 20,
- 2 students got scores in the interval [16.1, 16.3],
- 5 students got scores in the interval [11, 14.5].
The median is 14.35 (out of 20, so 71.75%).
- Here are some comments about the grading, and common mistakes that some made.
- Instead of giving a conditional probability when one was requested, some students tried to give the probability of what I guess may be called a
"conditional event" (which is not a recognized probability concept). In general, an event should not be defined using something of the form
statement a
given
statement b. (Instead of doing that, it'd be better to define events A and B, and give the conditional probability of A
given B.)
- Some lost points (or at least a fraction of a point) because they didn't provide adequate justification for their answers. (E.g., if you go
from the probability of an intersection of two events to the product of their probabilities, you should state that this is due to independence.
Or you might introduce part of a derivation by stating "Due to the independence of the Ai we have ... ")
- Define events, not probabilities. That is, don't define P(A). Just define A, and then it's clear what P(A) represents.
- Extra Ch. 3 Problems (not to be graded)
- HW 3 (due at class time on Feb. 24,
but grace period until 11:30 PM on 2/24)
- solutions for HW 3
- The 5 parts graded: 1, 2, 3, 4(c), 5(a).
- Here is a summary of the scores:
- 3 students got scores in the interval [19.5, 20],
- 3 students got scores in the interval [13.7, 16.8],
- 2 students got scores below 10.
The class median is 16.35 (out of 20, so 81.75%).
- Here are some comments about the grading.
- Some students are still abusing notation ... like having a union of probabilities. (The probability of a union is fine, but a union of
probabilities isn't defined ... so not sensible.) Also, P(A+B) is an abuse of notation.
- Some students used what I guess may be called
"conditional events" (which is not a recognized probability concept). In general, an event should not be defined using something of the form
statement a
given
statement b. (Instead of doing that, it'd be better to define events A and B, and give the conditional probability of A
given B.)
- I had to penalize some students for not giving adequate justification for some of their answers. You might think a problem is so simple that
no explanation is needed, but here's my take on the situation: some students who try to justify their answer give poor justifications (they get
the right answer, but the justification for it is faulty), and I take off for having improper justification ... and so I felt like I couldn't give full credit
to those who offered no explanation, or not adequate explanation, for their answer (because had they tried to justifiy their answers, I don't know if their
explanations would make sense or not). For part 4(c), a correct answer with no justification only earned 2 points.
- Something cannot be both an event and a random variable. E.g., if you define G as an event, then you shouldn't have things like G = 2.
And if X is a random variable, then P(X) doesn't make sense.
- Finally, please draw boxes around your final answers, and staple sheets together.
- Extra Ch. 3 and Ch. 4 Problems (not to be graded))
- HW 4 (due at class time on Mar. 2,
but grace period until 11:30 PM on 3/2)
- solutions for HW 4
- The 5 parts graded: 1, 3, 4, 5(b), 6.
- Here is a summary of the scores (for the 7 students who submitted solutions):
- 3 students got scores in the interval [16.5, 18],
- 3 students got scores in the interval [10.9, 14.3],
- 1 student got a score below 5.
The median is 14.3 (out of 20, so 71.5%).
- Here are some comments about the grading.
- Two students got the correct answer for Problem 1 (about the wolf and the sheep), but one student did not give any sensible justification for the answer, and the
other student gave a very complicated explanation (which I didn't fully understand) when a rather simple one is possible. So the high score for that problem was
only 2 out of 4 points.
- When you give a pmf as an answer, as a check, you should make sure that the sum of the probabilities for the possible outcomes equals 1.
- Extra Ch. 4 Problems (not to be graded)
- HW 5 (due at class time on Mar. 23,
but grace period until 11:30 PM on 3/23)
- solutions for HW 5
- The 5 parts graded: 2, 3(a), 3(b), 4(a), 4(d).
- Here is a summary of the scores:
- 3 students got scores in the interval [19.3, 19.9],
- 1 student got a score of 18,
- 3 students got scores in the interval [6.6, 13.2].
The median is 18 (out of 20, so 90%).
- Here are some comments about the grading.
- Some students wrote expressions like
P( A ) ∪ P( B )
instead of
P( A ∪ B )
... but a union of probabilities doesn't make sense.
- Some used X (or A) as both a random variable and an event in the same problem ... it can't be both!
- Sadly, some students didn't seem to realize that a negative probability or a probabilty greater than 1 isn't sensible. (If you give a probability that is
outside [0, 1] as your answer at this stage of the semester, you shouldn't expect to earn much partial credit.)
- Extra Ch. 4 and Ch. 5 Problems (not to be graded)
- HW 6 (due at class time on Mar. 30,
but grace period until 11:30 PM on 3/30)
- solutions for HW 6
- All 5 parts were graded
- Here is a summary of the scores:
- 1 student got a score of 19.8,
- 3 students got scores in the interval [14.25, 16.1],
- 4 students got scores in the interval [10.6, 13.3].
The median score is 13.775 (out of 20, so 68.875%).
- Extra Ch. 5 Problems (not to be graded)
- HW 7 (due at class time on Apr. 6,
but grace period until 10:30 PM on 4/6)
- solutions for HW 7
- The 5 parts graded: 1(a), 1(b), 2, 3, 4.
- Here is a summary of the scores:
- 3 students got scores in the interval [19.35, 20],
- 3 students got scores in the interval [14.6, 17.3],
- 2 students got scores in the interval [5.6, 10.65].
The class median is 18.325 (out of 20, so 91.625%).
- Here are some comments about the grading.
- Some students are still arriving at pdfs which take negative values, and cdfs which take values outside of the range [0, 1],
or which decrease instead of increase over the distribution's support. Also,
some arrive at a value for an expectation which is outside the range of values that the random can assume, and that's not sensible. So, it's obvious that some need to give
more consideration as to whether or not their answers are even sensible.
- Some students are not properly indicating the support when they give a pdf. Unless the density is positive for all real numbers, you need to indicate what
the support is in some way.
- Some students used random variables in their solutions without defining them.
- Additional Extra Ch. 5 Problems (not to be graded)
- HW 8 (due at class time on Apr. 13,
but grace period until 10:30 PM on 4/13)
- solutions for HW 8
- The 5 parts graded: 1, 2, 4, 6, 8(a).
- Here is a summary of the scores (for the 7 students who submitted a paper):
- 2 students got scores in the interval [19.8, 19.9],
- 2 students got scores in the interval [18.8, 19.1],
- 3 students got scores in the interval [15.3, 15.9].
The class median is 18.8 (out of 20, so 94%), of the 7 papers submitted.
- Here are some comments about the grading.
- Some of you are giving negative probabilities or probabilities larger than 1, which of course should be obviously wrong. Also, noting that a cdf value is a probability, a cdf should never be negative or exceed 1 (and for a continuous random variable, the cdf should increase from 0 to 1 over its support).
- Some people are giving too many digits when they report a probability involving a normal distribution. If you use interpolation, you should round to the nearest
thousandth, and also you should round if you use a normal approximation of a binomial distribution probability. (For the purposes of this course, round to the
nearest thousandth unless instructed otherwise (but sometimes even that reflects too much accuracy).)
However, don't round too much before the last step.
- Extra Ch. 6 Problems (not to be graded)
- HW 9 (due at class time on Apr. 20,
but grace period until 10:30 PM on 4/20)
- solutions for HW 9
- The 5 parts graded: 1(a), 1(b), 2, 3(a), 3(b).
- Here is a summary of the scores:
- 1 student got a score of 19.95,
- 5 students got scores in the interval [16.1, 17.9],
- 2 students got scores in the interval [12.2, 13.6].
The class median is 16.75 (out of 20, so 83.75%).
- Here are some comments about the grading.
- Some students are still failing to include an indicator function or otherwise clearly indicate the support of the distribution when giving a pmf or pdf.
- Some students gave pdfs for which the area under them was 0, infinite, or some other value which is clearly not sensible ... a pdf has to integrate to 1.
- Some students gave pdfs and/or cdfs which take negative values ... which cannot be correct.
- Some students gave cdfs that were equal to 0 outside of the support ... but cdfs are only 0 below the support, and above the support they equal 1.
- Recall that a cdf should increase from 0 to 1 on it's support (but if the support is of infinite length, then it may only asymptotically approach 0, or
asymptotically approach 1).
- If a pdf is expressed as a sum of two parts, say a(x) + b(x), on its support, then one needs to write (a(x) + b(x)) before multiplying by an indicator function.
If just b(x) is multiplied by an indicator function, then that indicator does not apply to a(x) ... the parentheses are needed in order for the indicator of the
support to apply to both a(x) and b(x).
- Additional Extra Ch. 6 Problems (not to be graded)
- HW 10 (due at class time on Apr. 27,
but grace period until 10:30 PM on 4/27)
- solutions for HW 10
- All 5 parts were graded.
- Here is a summary of the scores:
- 1 student got a score of 19.9,
- 3 students got scores in the interval [16, 17.1],
- 3 students got scores in the interval [10.15, 12.7].
The median score is 16 (out of 20, so 80%), out of the 7 papers submitted.
- Extra Ch. 7 Problems (not to be graded)
- HW 11 (due at class time on May 4,
but grace period until 10:30 PM on 5/4)
- Additional Extra Ch. 7 Problems (not to be graded)
- HW 12 (due at class time on May 11,
but grace period until 10:30 PM on 5/11)
Presentation of Homework Solutions
- Use paper which is approximately 8.5 inches by 11 inches.
- Staple (as opposed to paper clipping or "dog-earing") sheets together in upper left corner.
- Present solutions in proper order. (E.g., solution for Problem 1 should come before solution for Problem 2, and solution for part (a)
should come before solution for part (b).)
- Draw boxes around or highlight your final answers (or you can use a bold font if you type your work).
- Give just one answer for each part --- don't hedge by indicating that the answer is either this or that.
- Unless problems are truly trivial, provide adequate justification for your answers.
- Express numerical answers exactly, or else round to three significant digits. (The following numbers have three significant digits: 20.8, 0.208, and 0.00208.
The numbers 0.020 and 0.002 have only two and one significant digits.) If an approximation formula is used to obtain a numerical answer, don't indicate too much
accuracy by giving a lot of significant digits. (Often two significant digits are all that are warrented for an approximate answer, but follow any specific guidelines that I give in class or with the assignments.)
- Unless it's stated otherwise, don't express answers as messy expressions that need to be evaluated (e.g., involving binomial coefficients or integrals).
For things like probabilities and expected values, I generally want numerical answers. (For HW 1 I want each answer given in numerical form (either as a fraction or
in decimal form (possibly rounded to 3 significant digits (which is not necessarily the same thing as rounding to the nearest thousandth)).)
- Unless they are defined for you in the statement of a problem, be sure to clearly define any events and random variables that that you use.
For example,
for Problem 4 of HW 1 you might define G to be the event that all of the balls selected are green, and for Problem 2 of HW 2 you might define
B to be the event that all of the blue balls are in the same group.
(Just putting something like B = blue does not clearly define the event B.)
Late HW Policy (and Submission of Papers Outside of Class)
- You can turn in homework up until 11:30 PM on the night it is due.
(This grace period applies to all homework assignments this semester.)
- When dropping off late HW papers,
take them to my office in
Nguyen Engineering Building,
putting them under my door (my office is room 1706).
- If you fax your paper (to (703) 993-1700),
send me an e-mail indicating that you did, so that I can go fetch your paper from the department's fax machine.
- Never e-mail me your solutions ---
I want a paper copy that I can carry around and grade.
- If you fax your paper, or drop it off, and for whatever reason I don't
get it, then I won't give you any credit. So the best plan will be to
always turn in your paper to me in the classroom on the Tuesday that
it is due (but it is usually safe for you to put your paper under my
office door --- I don't know of any instances when students have done so
and I didn't get the paper).
Academic Integrity
While you can discuss homework problems with other students or me, you should not just copy
someone else's solution. After discussing the problems, you should go off by yourself and write up your own solutions (never looking at someone else's paper while
you do so). If I feel that some students are working too closely together I may give a warning. Then, if the problem persists, I may state that it will be an Honor
Code violation to discuss homework problems and share solutions with other students (and I will turn in suspected violators to the Honor Committee).