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

PRINT your name and student ID:

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

PRINT your name and student ID:

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

PRINT your name and student ID:

4. Said no more counting dollars,

EECS 70, Fall 2014, Final exam 4

PRINT your name and student ID:

5. We’ll be counting stars

EECS 70, Fall 2014, Final exam 5

PRINT your name and student ID:

6. And we’ll be counting bars!

(a) I see these balls

(b) Falling into bins.

EECS 70, Fall 2014, Final exam 6

PRINT your name and student ID:

7. Count the number of locations:

n+ k−1

EECS 70, Fall 2014, Final exam 7

PRINT your name and student ID:

8. Choose k and ye shall find.

EECS 70, Fall 2014, Final exam 8

PRINT your name and student ID:

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

PRINT your name and student ID:

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

PRINT your name and student ID:

(c) I just try the Improvement Lemma

EECS 70, Fall 2014, Final exam 11

PRINT your name and student ID:

11. I want to prove something

EECS 70, Fall 2014, Final exam 12

PRINT your name and student ID:

[Extra page. If you want the work on this page to be graded, make sure you tell us on the problem’s mainpage.]

EECS 70, Fall 2014, Final exam 13

PRINT your name and student ID:

12. By doing direct proof

EECS 70, Fall 2014, Final exam 14

PRINT your name and student ID:

[Extra page. If you want the work on this page to be graded, make sure you tell us on the problem’s mainpage.]

EECS 70, Fall 2014, Final exam 15

PRINT your name and student ID:

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

PRINT your name and student ID:

14. By counterexample!

Mark one: TRUE or FALSE.

EECS 70, Fall 2014, Final exam 17

PRINT your name and student ID:

15. I couldn’t lie, couldn’t lie, couldn’t lie...

Mark one: TRUE or FALSE.

EECS 70, Fall 2014, Final exam 18

PRINT your name and student ID:

16. Everything in 70 makes me feel alive.

Mark one: TRUE or FALSE.

EECS 70, Fall 2014, Final exam 19

PRINT your name and student ID:

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

PRINT your name and student ID:

(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

PRINT your name and student ID:

(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

PRINT your name and student ID:

[Extra page. If you want the work on this page to be graded, make sure you tell us on the problem’s mainpage.]

EECS 70, Fall 2014, Final exam 23

PRINT your name and student ID:

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

PRINT your name and student ID:

(b) (5 points) Learn them allFor the final

EECS 70, Fall 2014, Final exam 25

PRINT your name and student ID:

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

PRINT your name and student ID:

[Extra page. If you want the work on this page to be graded, make sure you tell us on the problem’s mainpage.]

EECS 70, Fall 2014, Final exam 27

PRINT your name and student ID:

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

PRINT your name and student ID:

(c) (10 points) Everything in 70 makes me feel alive.

EECS 70, Fall 2014, Final exam 29

PRINT your name and student ID:

[Extra page. If you want the work on this page to be graded, make sure you tell us on the problem’s mainpage.]

EECS 70, Fall 2014, Final exam 30

PRINT your name and student ID:

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

PRINT your name and student ID:

(c) (10 points) And we’ll be counting bars!

EECS 70, Fall 2014, Final exam 32

PRINT your name and student ID:

[Extra page. If you want the work on this page to be graded, make sure you tell us on the problem’s mainpage.]

EECS 70, Fall 2014, Final exam 33

PRINT your name and student ID:

[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