ECE 302: Probabilistic Methods in Electrical and Computer EngineeringFall 2019Instructor: Prof. A. R. Reibman

Exam 1

Fall 2019, T/Th 3:00-4:15pm(September 23, 2019)

This is a closed book exam with 8 multi-part problems. Neither calculators nor help sheets are

allowed.

Cheating will result in a zero on the exam and possibly failure of the class. Do not cheat!

Use of any electronics is considered cheating.

Put your name or initials on every page of the exam and turn in everything when time is up.

Write your answers in the boxes provided. We will be scanning the exams, so DO NOT WRITEON THE BACK of the pages!.

Name: PUID:

I certify that I have neither given nor received unauthorized aid on this exam.

Signature:

SOLUTIONS

Problem 1. (True/False: 4 points each, total 24 points)

For each of the following statements, determine which is valid.

If you show your reasoning you might get partial credit.Finding a counter-example might be helpful if the answer is FALSE.

(Note: if a statement is not always true, then it is FALSE.)

Clearly label each statement T or F in the box to the left of the problem.

(a) If X is a RV and FX(x) is its CDF, then P (X > x) = 1� FX(x) for all x.

(b) If X is a RV and fx(x) is its PDF, then fX(x) 1 for all x.

(c) P (A \B) = P (A) + P (B)� P (A [B) for any two sets A and B.

(d) E[(X � 1)3] = [E(X)� 1]

3

(e) If A, B, and C form a partition, then Ac, Bc

, and Ccare collectively exhaustive.

(f) A correct mathematical expression for “the probability that X is no more than 1

away from a constant c is at least 95%” is

P (|X � c| 1) � 0.95

2

g PIX x I P 2 2 39 1 PL XEx

L Fxix for any RV

Fex f xix of

k is a valid PDF

1

Add PLAVB and subtract p AnB frm bothsides to get a corollary

F E X 3 4 1

Eid BEA t Ek 1

TF Efx 3 3ELX ELX I

A UB S so surelyT A UB Vc S too

reshot underlined

The event of interest wordsAllthe7 event ofFFI interest

Problem 2. (12 points (6 points for (a); 4 points for (b)))One Blue and one Green die are rolled. Both dice are fair. Let A be the event the number on the

Green die is even, and let C be the event that the sum of the two rolls is equal to 6.

(a) What is P (C|A)?

(b) Are events A and C independent? Explain your reasoning.

(No points without a proper reason.)

(a)

(b)

3

S g b g green die b blue die

Is 369 be 6

A green die is even PIA Iz

C s 12,4 3,31 4,415,1 PCC 536

Anc 2,4 4,2 Plane 436

a Peart Yi 4 Fa'tb Need PCAA p A Pcc or equivalently

PCHA Pcc

49

No Plane Za t I PCA Pk

and PCHA f Pcc 5136

Problem 3. (12 points (6 points each part))An engineer is designing a system to detect whales in the ocean. Let the event W indicate a whale

is present, and let W cindicate there is no whale. Let the event D indicate the system decides a

whale is present, and let Dcindicate the system decides there is no whale.

(a) If the system decides wrong 5% of the time there actually is a whale, and 5% of the time

there is actually no whale, and the prior probability of there being a whale is P (W ) = 0.2,what is the probability that the system decides D?

(b) Let there be a cost for two possible outcomes: that of missing the whale, and that of saying

there’s a whale when there’s no whale. Let the cost of the first case be 1000 for each occur-

rence, and the cost of the second be 10. Assume there is no cost to a correct decision. Using

the result from part (a), what is the expected cost of this system each time it is used?

(a)

(b)

4

a decides wrong solo of time there is a whale

P D l w 0.05

decides wrong 5 of time there is no whale

PCD Iwa 0.05

Thin total probi PCD P DIN PCW PCDIwc p w1 0.05 o 2 t 0.05 l O

7 0.19 10.040.222,6

c cost 1000 if Dfw10 if Dn we0 else

E c 1000 PC Dn wa t lo Pl D n w

0.23

10.4

E c 1000 PC 151W PCW t 10 PCD Iwgp W1000 05 2 t 10 0.05 C 8 10 10.4

Problem 4. (12 points (6 points each part))A wallet contains two coins. One is a fair coin; the other has a probability of tails of 0.25. You

pick a coin at random and give the other coin to someone for safe keeping.

(a) You flip the coin you picked; what is the probability of getting a tails?

(b) You flip the coin you picked a second time. What is the probability of getting a tail on the

second flip, given that you got a tail on the first flip?

(a)

(b)

5

S

TntYu Tin Fair

MaiaI YieldingFIIails

TFa 1

PIT PCT I Fair PCFair PCT1 Fair P Faire

Iz Iz t t I 3 8

b P Ta IT PCT h Tz PLT Atl Fair PCFair 1

PLT Atl Fairc Pffair

ITE Ees o g3 8 E EE E8

Note you are using the same coin to512 You can also compute j

E ITPLT 1 Fair Pl Fair Yz f 8pet Ig If I

I z t t t I 2 IT318 Too Ez I

Problem 5. (12 points (6 points each part))

Suppose the PDF of X is given by

fX(x) =

⇢cx2 if 0 x 2

0 otherwise

(a) Find the value of the constant c.

(b) Find P (X 1).

(a)

(b)

6

a fifth dx I f cx2dx102

cogc 3 8

b PC Xfl f Ix dx lot

c 18

3 8

48

Problem 6. (12 points (6 points each part))

Suppose the PDF of X is given by

fX(x) =

⇢x+ �(x� 1/2)/2 if 0 x 1

0 otherwise

(a) Find the expected value of X: E(X).

(b) Find the variance of X: VAR(X)

(a)

(b)

7

recall sifting property

figtx six a dxg a

ELX xfxlx dx x xx'z8 x I Ddx

of x'dx If'x six E dx ICE

Iz t 144 7µm

Varix ELM EK

ELM fo X f lx1dx x'dxtfftigfx.ttlo Iz t I t too 318

7 12

varix E E Ei Tft Eh 4

I 8

8114T I544

54

Problem 7. (12 points (6 points each part))

Suppose the CDF of the random variable Y is given by

FY (y) =

8>>>><

>>>>:

0 for y < 0

y/4 for 0 y < 2

3/4 for 2 y < 3

y/4 for 3 y < 4

1 for y � 4

(a) Sketch the CDF. Is Y a continuous, discrete, or mixed random variable?

(b) Find P (1 Y 2).

Sketch:

Continuous, discrete, or mixed:

(b)

8

Fylyl

0042a.iii

mixed There is at least onejump and at least one interval

of continuity

Pll EYE 2 PC aye 2 PLY DFx 2 Fx l t 0

34 44

Problem 8. (6 points)

Let X be a random variable with mean µ and variance �2, and let Y = 2X2

+ 4X � 1.

Express E(Y ) in terms of µ and �.

E(Y ) =

9

ELY E 2 2 4 1

2 E N t 4 E X Ell

ZECK t 4M I

Varlx E XY E X

so EIX Varix ECX

62 1µL

So Ely 202 2M 4M I

Zo't 2h2 4µ I