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EECS 70 Discrete Mathematics and Probability TheoryFall 2014 Anant Sahai Final exam
Exam location: 1 Republic
PRINT your student ID: 00000000
PRINT AND SIGN your name: Tedder , Ryan(last) (first) (signature)
PRINT your Unix account login: cs70-rep
PRINT your discussion section and GSI (the one you attend): TimbalandName of the person to your left: Zach Filkins
Name of the person to your right: Brent Kutzle
Name of someone in front of you: Eddie Fisher
Name of someone behind you: Justin Bieber
Section 0: Pre-exam questions (3 points)1. Lately I’ve been, I’ve been losing sleep... (1 pt)
2. Dreaming about the final and probability. (2pts)
Do not turn this page until the proctor tells you to do so.
EECS 70, Fall 2014, Final exam 1
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SOME APPROXIMATIONS AND OTHER USEFUL TRICKS THAT MAY OR MAY NOT COME IN HANDY:
n!≈√
2πn(ne)n
(nk
)≤ (
nek)k lim
n→∞(1+
xn)n = ex
When x is small, ln(1+ x)≈ x
When x is small, (1+ x)n ≈ 1+nx
The Golden Rule of 70 (and Engineering generally) applies: if you can’t solve the problem in frontof you, state and solve a simpler one that captures at least some of its essence. You might get partialcredit for doing so, and maybe you’ll find yourself on a path to the solution.
Source: http://www.math.unb.ca/~knight/utility/NormTble.htm
EECS 70, Fall 2014, Final exam 2
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Section 1: Straightforward questions (30 points)You get one drop: do 5 out of the following 6 questions. Bonus for getting all six perfectly. No partial creditwill be given.
3. But Sahai, I’ve been, I’ve been studyin’ hard!
EECS 70, Fall 2014, Final exam 3
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4. Said no more counting dollars,
EECS 70, Fall 2014, Final exam 4
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5. We’ll be counting stars
EECS 70, Fall 2014, Final exam 5
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6. And we’ll be counting bars!
(a) I see these balls
(b) Falling into bins.
EECS 70, Fall 2014, Final exam 6
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7. Count the number of locations:
n+ k−1
EECS 70, Fall 2014, Final exam 7
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8. Choose k and ye shall find.
EECS 70, Fall 2014, Final exam 8
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Section 2: Additional straightforward questions (27 points)You get one drop: do 3 out of the following 4 questions. Bonus for getting all of them perfectly. Very littlepartial credit will be given.
9. Paired,
EECS 70, Fall 2014, Final exam 9
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10. but I’m still rogue.
(a) Guess it’s just not stable
(b) And I don’t think my proof is wrong
EECS 70, Fall 2014, Final exam 10
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(c) I just try the Improvement Lemma
EECS 70, Fall 2014, Final exam 11
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11. I want to prove something
EECS 70, Fall 2014, Final exam 12
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EECS 70, Fall 2014, Final exam 13
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12. By doing direct proof
EECS 70, Fall 2014, Final exam 14
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EECS 70, Fall 2014, Final exam 15
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Section 3: True/False (30 points)You get one drop: do 3 out of the following 4 questions. Bonus for getting all four correct.
For each question in this section, determine whether the given statement is TRUE or FALSE. If TRUE, provethe statement. If FALSE, provide a counterexample or otherwise disprove it.
13. And I show something so wrong
Mark one: TRUE or FALSE.
EECS 70, Fall 2014, Final exam 16
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14. By counterexample!
Mark one: TRUE or FALSE.
EECS 70, Fall 2014, Final exam 17
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15. I couldn’t lie, couldn’t lie, couldn’t lie...
Mark one: TRUE or FALSE.
EECS 70, Fall 2014, Final exam 18
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16. Everything in 70 makes me feel alive.
Mark one: TRUE or FALSE.
EECS 70, Fall 2014, Final exam 19
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Section 4: Free-form Problems (73 + 25 points)17. Lately I’ve been, I’ve been losing sleep (hey!) (25 points)
(a) (5 points) Dreaming about the final and probability.
EECS 70, Fall 2014, Final exam 20
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(b) (5 points) But Sahai, I’ve been, I’ve been studyin’ hard!Said no more counting dollarsWe’ll be counting stars!
(c) (5 points) Lately I’ve been, I’ve been losing sleep (hey!)
EECS 70, Fall 2014, Final exam 21
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(d) (5 points) Dreaming about the final and probability.
Said no more counting dollars.
(e) (5 points) We’ll be, we’ll be counting bars.
EECS 70, Fall 2014, Final exam 22
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EECS 70, Fall 2014, Final exam 23
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18. I try Markov, (15 points)
But it’s not too tight
(a) (10 points) Chebyshev is much better.Chernoff is just too advanced.
EECS 70, Fall 2014, Final exam 24
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(b) (5 points) Learn them allFor the final
EECS 70, Fall 2014, Final exam 25
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19. Encrypt, use RSA (15 points)
Share secrets, in GF(p)
And I don’t think the secret is found
Since the shares are still hidden.
Mark one: TRUE or FALSE.
If TRUE, prove the statement. If FALSE, provide a counterexample or otherwise disprove it.
EECS 70, Fall 2014, Final exam 26
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EECS 70, Fall 2014, Final exam 27
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20. I average my data (18 points)
By the expectationAnd I learn how they spread out
(a) (3 points) By computing the variance.
(b) (5 points) I couldn’t lie, couldn’t lie, couldn’t lie...
EECS 70, Fall 2014, Final exam 28
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(c) (10 points) Everything in 70 makes me feel alive.
EECS 70, Fall 2014, Final exam 29
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EECS 70, Fall 2014, Final exam 30
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21. (Optional) Lately I’ve been, I’ve been losing sleep (hey!) (25 points)
Dreaming about the final and probability.But Sahai, I’ve been, I’ve been studyin’ hard
(a) (5 points) Said no more counting dollars,
(b) (10 points) We’ll be counting stars
EECS 70, Fall 2014, Final exam 31
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(c) (10 points) And we’ll be counting bars!
EECS 70, Fall 2014, Final exam 32
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EECS 70, Fall 2014, Final exam 33
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[Doodle page! Draw us something if you want or give us suggestions or complaints.]
Almost done! Good luck, everyone!
EECS 70, Fall 2014, Final exam 34