ECE 302: Probabilistic Methods in Electrical and Computer Engineering

Instructor: Prof. A. R. Reibman

Past Exam Questions(Fall 2015, Spring 2016, Fall 2016, Fall 2017)

Chapters 3 and 4

Reibman(January 2019)

These form a collection of problems that have appeared in either Prof. Reibman’s real exams or“sample exams.” These can all be solved by applying the material we covered in class that appearsin Chapters 3 and 4 of our textbook.

The last 3 pages of this document will be provided to you as the last pages of the exam. This willbe all the formulas that will be available to you. The rest you must memorize.

Solution

solutions to 59 Gyand 66 70 only

Problem 56.

The peak temperature T in degrees Farenheit, on a July day in Antarctica is a Gaussian RV withvariance 225. With probabilty 1/2, the peak temperature T exceeds 10 degrees. What is P (T > 32),the probability the peak temperature is above freezing?

41

T is Gaussian w mean µ and variance 225

µ is not explicitly givenbut PCT 101 42

which means the median is 10 degrees

so µ 10

PCT 32 PC T 37

PLZ 2 where Z NN o D

plz 1.47

0.071

ft It

i

Problem 57. (5 points)

Let X be a Gaussian random variable with mean 52 and variance 4. What is the probability thatX > 57 given that X > 54?

(You may leave the (approximate) answer in terms of a ratio of integers.)

42

X N N 52,4

Z NN lo l if 2 X

P X 57 X 54 Pl X 57 A X 54

PIX S7syg

547 I I 57252

FIFEI Telsh OIL 512FIELD FED0.0062

07587

Problem 58. (15 points)

Two manufacturers label resistors as 100-ohm resistors, but in reality the actual resistence is arandom variable, X. For one company, X is Gaussian with mean 100 and variance 9. For thesecond company, X is Gaussian with mean 100 and variance 4. Equal numbers of resistors fromeach company are purchased and then mixed together.

(a) If you pick a resistor at random, what is the probability its resistance is between 97 and 100?

(b) If you pick a resistor with resistance X = 97, what is the probability it was made by Company2?

Note: You may use (and detach) the �-function table on the last page of the exam.

43

a we will use theorem of total probability to solvethis

P 97 LX c 100 P 97 X doo G P G

t PC97 a X 400 Cz PCG

solve each piece separately

Pl 97 X a 100 G p 9 00 a z e 10031001

P f l c Z so where 2 NN oD

Et o I l t

Pl 97 X a 100 4 P 97 00a z e

00 10021Pf Z co Ef o IEC 312

combining PC 97 x c too OI co I E l D I tf 40,38725

b Use Bayes Rule

p Cz I x 97 ftfx 97

Problem 59. (Multiple choice: 6 points)If X is a uniformly distributed RV on the interval [0, 1], then the pdf of the RV Y = 5X is

(a)

fY (y) =

⇢1/5 for 0 y 10 otherwise

(b)

fY (y) =

⇢5 for 0 y 1/50 otherwise

(c)

fY (y) =

⇢1/5 for 0 y 50 otherwise

(d)

fY (y) =

⇢5 for 0 y 10 otherwise

(e)

fY (y) =

⇢1 for 0 y 10 otherwise

44

not a pdf

wrongheightwrong bounds

not a pdfwrongheightfounds

Given fax

y f o II

Rea f ly ya f tab if Y aXxb

here 4 5 X SO45 of YES

fyly gtfx Is o else

check Iffy g dy I r

Problem 60. (15 points)Let X be a voltage input to a rectifier, and let X be a continuous uniform random variable on theinterval [�1, 1]. The rectifier output is a random variable Y , where

Y = g(X) =

⇢0 X < 0X X � 0

Find and sketch the PDF of Y , fY (y).

45

Two solutionapproaches

Approach 1 5

9471A

Two steps y

Find Fycy1

differentiate togeffyly

In step there are 3 casesfxgqy.cz

y Lo y 0 and y o

case 1 yo l l

Fyly PlYEy o since Yis never lessthan 0

case 2 y O

Fy lo p Yeo P Xiao fxlx dx dx L

case 3 y O Fyly

Fyly PLY Ey Play Fx y Yet0TDifferentiate

y 20

fishy

is go f p EoYz o yal

Problem 60. (15 points)Let X be a voltage input to a rectifier, and let X be a continuous uniform random variable on theinterval [�1, 1]. The rectifier output is a random variable Y , where

Y = g(X) =

⇢0 X < 0X X � 0

Find and sketch the PDF of Y , fY (y).

45

Two solutionapproaches

Approach 2 5 gtxyyet1Every value of

X

gets mapped intothe

same value 4 0

So 4 0 happens halffxl in

the timex

IS y at zero l

The positive values of X are unchanged

sothere's no change to the density there

so fy'Dµ

y

Problem 61. (20 points)Suppose self-driving cars have become a reality, in which you, personally, have no control over thespeed. You program your car with a destination, and the car drives at a constant speed throughoutthe trip. This speed is chosen by the car before the trip starts, and the speed is held constantthroughout the trip.

Suppose you “drive” (in your self-driving car) from West Lafayette, IN to Louisville KY at a con-stant speed that is uniformly distributed between 30 and 60 miles per hour (mph). The trip isexactly 180 miles. What is the PDF of the duration of the trip?

Another description of the problem that is completely equivalent:

Suppose X and Y are random variables, where X is the speed and Y = 180/X is the duration. IfX is uniformly distributed between [30, 60], what is the PDF of Y ?

(Hint: first decide: is X a continuous or discrete RV?)

46

RecallUse Z stepprocess f boo

30 6

Find Fy y else

differentiate wrt y

Fyly p y e y p 8 Ey Pf8yd ex

I P X 2 I Fx Hf

f ly dayFyly fx Ig Ig Igfx G f 18 f Eg

Y'hanginthia

Range for Y when X 30 y 6 when 11 60 Y 3

Now substitute fx x to get

fyly Foto II II

Problem 62.

Let Y = 2X + 3. Find the PDF of Y if X is a uniform RV on [�1, 2].

47

when Y aXtb fyly af f tabhere a Z b 3

fyly zf KIthis is

O when

x If s l

ory

D 2

so it's zero for yalor for y 37

The problem statement tellsus f x texe

0 else

so fy yto Ky's

0 else and yes

fyly dy

Problem 63.

If X is a positive random variable with density fX(x), find the density of +pX. Apply this to find

the pdf of the length of a side of a square when the area of the square is uniformly distributed in[a, b].

48

let Y glx tixif X is the area of a square

faxthen Y is the length of its

side

fx x 4lb a at b

a b o else

we know fy y o when y 0

when y o Fy y PCYey P Tx Ey

p x e y Fx y

fy ly Ey Fyly f y Fylyf ly2 zig using the charina

substituting fx x from the problem statement

fy ly awhen Va y Erb

O fylyelse

g

Problem 64. (Multiple choice: 5 points)

Let Y = X1/3, and let X be a continuous random variable that is uniformly distributed on theinterval [�1, 8]. What is the PDF of Y ?

(Note: if you show your work you may get partial credit.)

(a)

fY (y) =

⇢3y2/513 for � 1 < y < 80 otherwise

(b)

fY (y) =

⇢y2/3 for � 1 < y < 20 otherwise

(c)

fY (y) =

⇢1/3 for � 1 < y < 20 otherwise

(d)

fY (y) =

⇢1/9 for � 1 < y < 80 otherwise

(e)

fY (y) =

⇢3y2 for � 1 < y < 20 otherwise

(f) None of the above.

49

13y X

wrong range

since Y glXis nonlinear theuniform pdf is

wrongrangealtered

not a pdf

2 step processf tx 1a

T Fy ly P yay8 p x

3 Egp X ey3 Ely

differentiate fy ly Fy Fyly f Ly3 3y4by chain rule

f 135 in the Proffingeonly

range y2 3

when X l Y 1 and when X 8 4 2So f ez answer

Problem 66. (5 points)Given the CDF of X,

FX(x) =

⇢1� (2/x)2 for x > 20 otherwise

what is the PDF of Y = X2, fY (y).

51

mistakes to avoid

if 4 112 Fyg Fxtxy fy y µBoth are wrong

There are at least 2 approacheswith a common start

Fyly PLY Ey p 12

Ey PCx cryonly need the positive part because X alwayspositiveFx Ty

Appwatchl substitute Fx Ry and then differentia

7

Fuss5

f Is e

fyly Fy Fyly FyL 45 14y wheny.sc

AppTach2i differentiate Fyly usingchain rule

and Fx Ix to get fx x and substitute

fyly Fy Ex Ry fury Ey Ry fx Cry Tryf Xx adz Fxix Fx 2 8

3 when x 2

combine fyly Ry fry crypt 4452547

Problem 67. (15 points)Suppose a continuous random variable X is input to a quantizer to create Y , where

Y = g(X) =

8<

:

�2 for x < �10 for � 1 x < 1+2 for x � 1

(a) Express the PDF of the random variable Y in terms of the PDF fX(x) of the input X, usingan expression that will be true for any fX(x). You may find it useful to sketch g(X).

(b) Suppose fX(x) is a uniform distribution between [�3, 5]. What is E(Y )?

52

gtx _y 12

istTxy2

a for any PDF the event Xc I

corresponds to the event 4 2

so fyly will have a 8 function y 2

with mass Plx a 1

similarly fyly will havea 8 function y 2

with mass pix DBecause 4 211 for 1 Exel for 2 y Z

fy ly Efx 912 by the linear shortcut

sofy y

Fx C 1 Sly 12 y z

fx 9 2 zeya 2

I Eidsly 2 y 12

0 else

Problem 67. (15 points)Suppose a continuous random variable X is input to a quantizer to create Y , where

Y = g(X) =

8<

:

�2 for x < �10 for � 1 x < 1+2 for x � 1

(a) Express the PDF of the random variable Y in terms of the PDF fX(x) of the input X, usingan expression that will be true for any fX(x). You may find it useful to sketch g(X).

(b) Suppose fX(x) is a uniform distribution between [�3, 5]. What is E(Y )?

52

94 _y z f lX

f tif ti X jx

y

b fy yFx C 1 Sly 12 y 2

f 9 2 zeya 2

total rt seise

for the specific pdfFx C 1 too 3 D ty

1 Fx l too 4 k

so ELY t L 2 Ela EffydyE It to TI Iz 4 4

fatty3

Problem 68. (15 points)Let X be a continuous random variable with probability density function

fX(x) =

⇢x/2 for 0 x 20 otherwise

X is input to a system with output Y = g(X), where

g(X) =

8<

:

�2(X + 1) for X < �10 for � 1 X < 1

+2(X � 1) for X � 1

Find the PDF of the output random variable Y .

53

gtxof

P

Two sleep process fxtx1 Find Fyly2 differentiate toget fyly I

3 cases yo yo y 0 by examining gtx

1 0 Fy y _0 because y can never beless than 0

y o Fylo P Yeo Plot XE 1 fo f lx dx

fo Handy 14 a jump in the

y o This paisrticular input X EDF y 0never less than zero so

this simplifies things a bitFyly P Yey play Dey just use the

differentiatePIX 1 Fx z y

3rd pieceof g

y co

fyly Fyly Iggy I 94Sly y o

i ii

Problem 69. (15 points)Let X be the voltage output from a microphone which is uniform on the interval from [�5, 5]. ThenX is input to a limiter circuit, with cut-o↵ ±4. Thus the output of the limiter Y is given by

Y = g(X) =

8<

:

�4 for X < �4X for � 4 X 44 for X > 4

Find and sketch the PDF of Y , fY (y).

54

glx

3 regions of interest

ye 4

yay c4y 4

Y can never be less than 4 faxor greater than 4

so fyly 0 for ya 4 5and y 4

for y 4 exactly PLY 4 PEIXE 4 To

for y 14 exactly PLY _4 PC 4 Exes tobecause

for 4cg c4 fy y fxly glx hasslope 1 in

Sofyly 819 4 5 4 this region410 419 4 fo fyly yo Yoo

Sly4

y 4

Problem 70. (5 points)Let X be a random variable with probability density function

fX(x) =

⇢2/x3 for x > 10 otherwise

Let Y = g(X) = min(X, 10). (This means that Y is either X or 10, whichever is less.) What is theexpected value of Y ?

55

NOTE this is the solution whenY maxlx.io not y min X lo as

stated

Eaxma LXgtx

break into 2 pieces g to

I XE 10 and x 10Tox

Eut Ethpix

4 quantities

P X Eto fxlx dx I PIX 10oo

Pl X lo fo fxlx dx2x 34 Too

El Yl X210 10oo

El Yl X 10 ftp.YY odx iooff2x2dx

100 I x 1,9 21 20 waistbiggerthan

so ELY 9 lo t lot Zoahessense

Problem 70. (5 points)Let X be a random variable with probability density function

fX(x) =

⇢2/x3 for x > 10 otherwise

Let Y = g(X) = min(X, 10). (This means that Y is either X or 10, whichever is less.) What is theexpected value of Y ?

55

This is the correct solution

gtx

break salaries

Eun Ethpix

x

4 quantities

P X Eto fax dx I PIX o

P X lo fax dx 2x 34 Foo

El yl Xero fYx dx Fg dxmuchless

than 0too E x I Fall to

finaleEl yl X 10 10 t

so ELY Hoo E to do 1stto HII