AAOC+C111(MATH+F113)

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BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI

HYDERABAD CAMPUS

SECOND SEMESTER 2012-2013

AAOC C111/MATH F113: PROBABILITY AND STATISTICS

COMPREHENSIVE EXAMINATION -PART A (CLOSED BOO!

D"#$: 03 MAY 2013 M"%& M"'): 30

D"*: FRIDAY T+$: &00 A&M& #. 10&00 A&M&

ID NO NAME:

:

NOTE: There are 10 questions in this paper. Each question below has four possible choices. Pic the choice

that is the !ost appropriate one an" put in the bo# as A$ % etc. Each correct choice fetches &ou 3 !ars

an" a wron' choice −1 !ar. A* "#$'"#+./.$''+#+ + 4$ 5.)+6$'$6 ") '. "6 ""'6$6

-1 "'& (ower case answers will also be consi"ere" as wron' answers.

z0.1=1.28,

z0.05=1.645,

z0.025=1.96,

z0.01=2.33

)1. If the chances of A hittin' a tar'et is 3 ti!es out of 4 an" that of B hittin' is

4 ti!es out of 5, an" that of C hittin' is 5 ti!es out of 6, an" if A ,B ,∧C

fire at the tar'et alternati*el& onl& once$ then the probabilit& that the tar'et will be hit e#actl&

twice will be +assu!e in"epen"ence,

+A,

19

24 +%,

47

120 +-,

23

30 +D,

15

22

). (et F ( x )=0 if x

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)2 (et X

be a ran"o! *ariable ha*in' Poisson "istribution with para!eter λ3/. Then P[ X = x ]

is

lar'est when x

is

+A, 2∨3 +%, 3∨4 +-, 1 +D, or no x

)4. The annual rainfall in a cit& is nor!all& "istribute" with !ean /0 inches an" s.". inches. If

c inches of rain is e#cee"e" +annuall&, 54 percent of the ti!e$ then c is

+A,33.29

+%,33.92

+-,26.71

+D,26.18

)6 If X ,Y are in"epen"ent ran"o! *ariables each ha*in' e#ponential "istribution with

para!eters α , β respecti*el&$ then the ran"o! *ariable K =min { X ,Y } will be

+A, a!!a with para!eters α , β +%, E#ponential with para!eter α + β

+-, E#ponential with para!eter

¿, β }min¿

+D, E#ponential with para!eter αβ

)7 If E [ X ]=2,Var X =5, E [Y ]=−1,VarY =3 an" Cov ( X ,Y )=2,

then Var (2 X −3Y +4 ) is

+A, 1 +%, 4 +-, 6/ +D, 23

)8 9ohn an" i!on ha*e a'ree" to !eet for lunch between 0.00 P.M.+i.e. 1.00 noon, an" 1.00

P.M. The arri*al ti!es of both are in"epen"ent ran"o! *ariables ha*in' unifor! "istribution o*er

+0$ 1,. Then the e#pecte" a!ount of ti!e +in hours, that the one who arri*es first !ust wait for

the other person is

(A) 1

3 +%,1

4 +-,3

4 +D,1

6

)5. The burnin' ti!e of a rocet +!easure" in secon"s, can be thou'ht of as a ran"o! *ariable ha*in'

unifor! "istribution on the inter*al (0,b). 10 rocets &iel"e" the followin' burnin'

ti!es: 8,7,10,9,7,12,11,7,9,10 . Then an unbiase" esti!ate of b will be

+A,12

+%,9

+-, 24 +D,18

)10. A re'ression line =b

0+b

1 x

was fitte" +b& the !etho" of least squares, to the

n "ata points ( x1, 1 ) ,( x2 , 2 ) ,!, ( xn , n) . If now each x i is increase" b& $ then

A

C

B

D

A

D

C

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+A, %othb

0,∧b

1 will chan'e +%, %othb0

,∧b1 re!ain the sa!e

+-,b1 re!ains the sa!e but

b0 will chan'e +D,

b0 re!ains the sa!e but

b1 will chan'e

* * *

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI

HYDERABAD CAMPUS



COMPREHENSIVE EXAMINATION -PART A(CLOSED BOO!

D"#$: 03 MAY 2013 M"%& M"'): 30

D"*: FRIDAY T+$: &00 A&M& #. 10&00 A&M&

ID NO NAME:

:

NOTE: There are 10 questions in this paper. Each question below has four possible choices. Pic the choice

that is the !ost appropriate one an" put in the bo# as A$ % etc. Each correct choice fetches &ou 3 !ars

an" a wron' choice −1 !ar. A* "#$'"#+./.$''+#+ + 4$ 5.)+6$'$6 ") '. "6 ""'6$6

-1 "'& (ower case answers will also be consi"ere" as wron' answers.

z0.1=1.28, z

0.05=1.645, z

0.025=1.96, z0.01=2.33

)1. If the chances of A hittin' a tar'et is 3 ti!es out of

4 an" that of B hittin' is

4 ti!es out of 5, an" that ofC

hittin' is 5 ti!es out of 6, an" if A ,B ,∧C

A

2 0 1 2 P S H

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fire at the tar'et alternati*el& onl& once$ then the probabilit& that the tar'et will be hit e#actl&

twice will be +assu!e in"epen"ence,

+A, 47

120 +%, 19

24 +-, 23

30 +D, 15

22

). (et F ( x )=0 if x

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)7 If E [ X ]=2,Var X =5, E [Y ]=−1,VarY =3 an" Cov ( X , Y )=2,

then Var (2 X −3Y +4 ) is

+A, / +%, 1 +-, 6/ +D, 4

)8 9ohn an" i!on ha*e a'ree" to !eet for lunch between 0.00 P.M.+i.e. 1.00 noon, an" 1.00

P.M. The arri*al ti!es of both are in"epen"ent ran"o! *ariables ha*in' unifor! "istribution o*er+0$ 1,. Then the e#pecte" a!ount of ti!e +in hours, that the one who arri*es first !ust wait for

the other person is

(B) 1

6 +%,1

3 +-,3

4 +D,1

4

)5. The burnin' ti!e of a rocet +!easure" in secon"s, can be thou'ht of as a ran"o! *ariable ha*in'

unifor! "istribution on the inter*al (0,b). 10 rocets &iel"e" the followin' burnin'

ti!es: 8,7,10,9,7,12,11,7,9,10 . Then an unbiase" esti!ate of b will be

+A, 9 +%, 12 +-, 18 +D, 24

)10. A re'ression line =b

0+b

1 x

was fitte" +b& the !etho" of least squares, to the

n "ata points ( x1, 1 ) ,( x2 , 2 ) ,!, ( xn , n) . If now each x i is increase" b& $ then

+A, %oth

b0,∧b

1

re!ain the sa!e +%, %oth

b0,∧b

1

will chan'e

+-,b

0 re!ains the sa!e butb

1 will chan'e +D,b1 re!ains the sa!e but

b0 will chan'e

* * *

A

B

C

D

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BIRLA INSTITUTE OF TECHNOLOGY 7 SCIENCE, PILANI

HYDERABAD CAMPUS

SECOND SEMESTER, 2012-2013

COMPREHENSIVE EXAMINATION

PART-B

AAOC C111/MATH F 113: PROBABILITY AND STATISTICS

DATE: 03-08-2013(FRIDAY! T+$: 120 M+& M"%& M"'): 80 "ote :# 0.0179=2.1,# 0.025=1.96,# 0.2389=0.707,# 0.0571=1.581, $ 0.025

2 =5.023, $ 0.0502 =3.841.

%rite o&r 'robabi(itva(&e)&'to fo&r*ecima( '(ace)∧)im'(if .

1.a, or a certain binar& channel$ the probabilit& that a trans!itte" ;0< is correctl& recei*e" as ;0< is

0.52 an" the probabilit& that a trans!itte" ;1< was recei*e" as ;1< is 0.51. urther$ the probabilit& of

trans!ittin' a ;0< is 0.24. If a si'nal is sent$ what is the probabilit& of 'ettin' an error= >2?

b, In a super !aret$ a case contains 1 -oca@-ola bottles$ in which / of the! are ha*in' e#pire"

"ate. A sa!ple of 2 bottles is ran"o!l& selecte" fro! the case. in" the !ean of $ the nu!ber of

bottles ha*in' the e#pir& "ate in the sa!ple usin' the probabilit& "istribution of $ an" also

calculate the theoretical !ean. >2?

a, An auto!obile pro"uction line turns out about 100 cars a "a&$ but "e*iations occur "ue to !an&

causes. The pro"uction in a "a& is !ore accuratel& "escribe" b& the probabilit& "istribution 'i*en

belowB

% 8 9 ; 100 101 102 103 10< 108

=(%! &03 &08 &0 &10 &18 &20 &18 &10 &0 &08 &03

inishe" cars are transporte" across the ba&$ at the en" of each "a&$ b& a ferr&. If the ferr& has space

for onl& 101 cars$ what will be the a*era'e nu!ber of cars waitin' to be shippe" in the first wee of

April +1st@7th, an" what will be the a*era'e nu!ber of e!pt& spaces on the ferr& "urin' that perio".

Cse the followin' ran"o! nu!bers 2$ 76$ 27$ 58$ 0/$ 67$ 1/ an" the tabular hea"in's for

si!ulation in the 'i*en or"er. >6?

Cse the followin' hea"in's: Da&$ Nu!ber of cars waitin' for transportation$ Fan"o! nu!ber$

Nu!ber of cars pro"uce"$ Nu!ber of cars ferrie" + inclu"in' the pre*ious "a& waitin' cars,$

Nu!ber of cars not ferrie"$ Nu!ber of e!pt& spaces on ferr&. Assu!e initiall& there is no

pro"uction.

b, If is a nor!al ran"o! *ariable with !ean 7 an" *ariance 2 then fin" the *alue of

P +14./62 G + H 7,G 0.054,= >/?

/. a, Two nu!bers are chosen in"epen"entl& an" at ran"o! fro! the inter*al +0$ 1,. hat is the

probabilit& that the two nu!bers "iff ers b& !ore than1

2+

>2?

b, (et +$ J, be "istribute" unifor!l& on the circular "is centere" at +0$ 0, with ra"ius2√ . in"

the !ar'inal "ensit& function of = >2?

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2. To test the h&pothesis that a coin is fair$ the followin' "ecision rules are a"opte". +i, Accept the

h&pothesis if the nu!ber of hea"s in 100 tosses is between 20 an" 60 both inclusi*e. +ii, reKect the

h&pothesis otherwise. >2L/?

a, in" the probabilit& of reKectin' the h&pothesis when it is actuall& correct.

b, hat conclusions woul" &ou "raw if the sa!ple of 100 tosses &iel"e" 4/ hea"s= -oul" &ou be

wron' in &our conclusion= If so$ what t&pe of error it coul" be=

4. A ran"o! sa!ple of 40 !athe!atics 'ra"es out of a total of 1000 showe" a !ean of 74 an" a

stan"ar" "e*iation of 10.

a. hat are the 54 confi"ence li!its for the !ean of the 1000 'ra"es= >/?

b. ith what "e'ree of confi"ence coul" we sa& that the !ean of all 1000 'ra"es is74 1±

=

>/?

6. The Koint "ensit& of the ran"o! *ariables an" J is

+ ,$ 0 1$ 0 1

+ $ , /

0 XY

x y x y f x y

elsewhere

+ ≤ ≤ ≤ ≤=

in" for this "istribution. in" also the cur*e of re'ression of J on . >7?

7. A process for pro"ucin' *in&l floor co*erin' has been stable for a lon' perio" of ti!e an" the

surface har"ness !easure!ent of the floorin' pro"uce" has a !ean 4.2. A secon" shift has been

hire" an" traine" an" their pro"uction nee"s to be !onitore". -onsi"er testin' the h&pothesis 0:

34.2 a'ainst 1: Q4.2. A ran"o! sa!ple of har"ness !easure!ents is !a"e of n320 speci!ens pro"uce" b& the secon" shift. If the !ean of the sa!ple is 4.1 an" stan"ar" "e*iation is 1.$ what

can be conclu"e" at 0.04 le*el of si'nificance= +Cse P@Ralue test, >4?

SSSSSS

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BIRLA INSTITUTE OF TECHNOLOGY 7 SCIENCE, PILANI

HYDERABAD CAMPUS

PROBABILITY AND STATISTICS

COMPREHENSIVE EXAMINATION

PART-B-SOLUTIONS

1 a, (et A be the e*ent of trans!ittin' ;02M!

0W+#,G0.0/ 354

0.0/W+#,G0.0

8

356

0.08W+#,G0.1

4

357

0.14W+#,G0.

4

358

0.4W+#,G0.2

0

355

0.20W+#,G0.6

0

3100

0.60W+#,G0.7

4

3101

0.74W+#,G0.8

4

310

0.84W+#,G0.5

310/

0.5W+#,G0.5

7

3102

0.57W+#,G1 3104

# 0 1 /

P+#, 0.424 0.4050 0.181 0.0181

1 / 2 4 6 7

1 0

2

58 58 0 /

0 7

6

10

10

1

1 0

/ 1 27

100

101

0 0

2 0 5

8

10

4

10

1

2 0

4 2 0

/

56 10

0

0 1

6 0 6

7

10

1

10

1

0 0

7 0 1

/

57 57 0 2

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11/2(1, 1)

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A*era'e nu!ber of cars waitin': 47 (1M! A*era'e nu!ber of e!pt& spaces: 87(1M!

b

¿ P [ 15.36420 60? >/5.4 60.4? P X P X ≤ ≤ = ≤ ≤

3

/5.4 40 60.4 40

4 4 P Z

− − ≤ ≤

+ince 3np340 an" X3np+1@p,34, @2M

3>.1?@>@.1? 30.562

+a, Y3P>FeKect the h&pothesis e*en thou'h it is true?

3P>FeKect the h&pothesis when p30.4?

[email protected]

30.0/48. @2M

+b, If 3 4/$ then accept the h&pothesis. Jes$ T&pe ZII error. @3M

4. +a, i*en

74$ 10$ 40 X s n= = =

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1. -usto!ers arri*e at a ban in a Poisson fashion at an a*era'e rate of 10 per hour. in" the probabilit& that

the ban staff ha*e to wait for not !ore than 6 !inutes +fro! the ti!e of ban2?

/. (et X be a continuous ran"o! *ariable with "ensit& f ( x )=e− x, x>0 +an" ¿0 otherwise,. (et

Y = 1

1+ X 2.

Obtain the "ensit& of the ran"o! *ariable Y . >8?

2. (et X be a continuous ran"o! *ariable with "ensit& f ( x )={ x0

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TEST 2 (OPEN BOO! (S.#+.)!

D"#$ : 30-03-2013 M"%& M"'): 90

D"*: SATURDAY T+$ : 11&00 A&M& #. 12&00 N..

1. -usto!ers arri*e at a ban in a Poisson fashion at an a*era'e rate of 10 per hour. in" the probabilit&

that the ban staff ha*e to wait for not !ore than 6 !inutes +fro! the ti!e of ban2?

olution: X therefore has a ❑2

"istribution with 0 "e'rees of free"o!.

ro! Table _R Pa'e 656$ P [8.26≤ X ≤31.4 ]=0.950−0.010=0.940

/. (et X be a continuous ran"o! *ariable with "ensit& f ( x )=e− x, x>0 +an" ¿0 otherwise,. (et

Y = 1

1+ X 2 Obtain the "ensit& of the ran"o! *ariableY .

olution: Y is a "ecreasin' function of X . ence the "ensit& of Y is

( )=f ( x )|*x*|=e− x 1

2 2√ 1 −1

=e−√ 1 −1 1

2 3/2

√ 1− ,0

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An" we 'et x=1.283

4. The two@"i!ensional ran"o! *ariable ( X , Y ) has a unifor! "istribution o*er the square with *ertices

(1, 0 ) , (0, 1 ) , (−1, 0 ) an" (0,−1 ) . (et / = X +Y ,V =Y − X . in" the Koint "ensit& (& ,v ) of

the two@"i!ensional ran"o! *ariable (/ ,V ) . ence fin" the "ensit& of the ran"o! *ariable

/ = X +Y . >1?

olution: F XY ( x , )=

1

2 o*er the square

The 9acobian 2 =|3 x

3&

3 x

3v

3

3&

3

3v|=|1/2 −1/21/2 1/2 |=1/2

ence/V (& ,v )=

1

4,−1

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6. A s&!!etric "ie is thrown /00 ti!es. (et X be the nu!ber of 4

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Poisson fashion at an a*era'e of 40 per "a&. or a 'i*en "a&$ fin" the probabilit& that at least two calls

will be recei*e" b& 5.14 A.M. >10?

. ro! e#perience$ Mr. Fa! has foun" that the low bi" on a construction Kob can be re'ar"e" as a ran"o!

*ariable X ha*in' unifor! "ensit& f ( x )={ 23C , C 2 8?

6. ow !an& ti!es "o we ha*e to flip a balance" coin to be able to assert with a probabilit& at !ost 0.01

that the "ifference between the proportion of tails an" 0.4 will be at least 0.02=

>10?

* * *

BITS- PILANI HYDERABAD CAMPUS

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TEST 2 MAE-UP(OPEN BOO! (S.#+.) 7 M"'+ S5?$$!

D"#$ : 1;-010?

olution:¿ P¿ at least calls are recei*e" between 5.00 A.M$ an" 5.14 A.M. ¿ +,

¿

¿1− P¿ at !ost 1 call is recei*e" between 5.00 A.M. an" 5.14 A.M. ¿ +,

¿1−e−7548 (1+ 7548 )=0.4629 +/L/,

. ro! e#perience$ Mr. Fa! has foun" that the low bi" on a construction Kob can be re'ar"e" as a ran"o!

*ariable X ha*in' unifor! "ensit& f ( x )={

23C , C

2 (1+ 8 100 )C )=2c−(1+ 8 100 )C

2C −C

2

=2

3(1−

8

100) +L,

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>10?

* * *

BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANIHYDERABAD CAMPUS

SECOND SEMESTER, 2012-2013AAOC C111/MATH F 113: PROBABILITY AND STATISTICS

Te! - 1 "C#$e% B$$' DATE: 2(-02-2013"M$)%+' Te: .0 M) M M: .0

Note: There are six questions. Anser a!! questions ith "usti#$ation %& $!ear!&'e#nin eent(s)/ran'o aria%!e(s). +ars -or ea$h question are ien at the en' o- the question.

1. In answerin' a question on a !ultiple@choice test$ a stu"ent either nows the answer or he

'uesses. (et 0.6 be the probabilit& that he nows the answer an" 0.2 the probabilit& that he 'uesses.

Assu!e that a stu"ent who 'uesses at the answer will be correct with 14$ where 4 is the nu!ber of

!ultiple@choice alternati*es. hat is the probabilit& that a stu"ent new the answer to a question

'i*en that he answere" it correctl&= @;M

. uppose that a cafeteria purchases 4 cartons of si! !il at the wholesale price of Fs 1.00 per

carton an" retails the !il at Fs 12.00 per carton. After the e#piration "ate$ the unsol" !il is

re!o*e" fro! the shelf an" the cafeteria recei*es a cre"it fro! the "istributor equal to three@fourths

# 0 1 / 2 4

f+#, 114 14 14 a14 214 /14

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of the wholesale price. If the probabilit& "istribution of the ran"o! *ariable $ the nu!ber of

cartons that are sol" fro! this lot is

in" the E#pecte" Profit for the -afeteria. @M

/. A ran"o! *ariable $ which represents the wei'ht +in ounces, of an article$ has the Probabilit&

"ensit& function is 'i*en b&

f ( x )={ ( x−8 ) ,for 8≤ x

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BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI

HYDERABAD CAMPUSSECOND SEMESTER, 2012-2013

AAOC C111/MATH F 113: PROBABILITY AND STATISTICSDATE: 1-0-2013 Te! - 1 MA4E UP"C#$e% B$$' Te: .0)DAY: THURSDAYM M: .0

Note: There are six questions. Anser a!! questions ith "usti#$ation %& $!ear!&'e#nin eent(s)/ran'o aria%!e(s). +ars -or ea$h question are ien at the en' o- the question.

1& The su! of two non@ne'ati*e quantities is equal to n. in" the probabilit& that their pro"uct is not

less than ̀ ti!es their 'reatest pro"uct. >8M?

2& A consi'n!ent of 14 recor" pla&ers contains 2 "efecti*es. The recor" pla&ers are selecte" at ran"o!

one b& one$an" e#a!ine". Those e#a!ine" are not put bac. hat is the probabilit& that the 5th one

e#a!ine" is the last "efecti*e. >8M?

3& A Piece of !echanis! consists of 11 co!ponents$ 4 t&pe A$ / or t&pe %$ of t&pe - an" 1 of t&pe D.

The probabilit& that an& particular co!ponent will function for a perio" of 2 hours fro! the

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co!!ence!ent of operations without breain' "own is in"epen"ent of whether or not an& other

co!ponent breas "own "urin' that perio" an" can be obtaine" fro! the followin' table:

-o!ponent A % - D

Probabilit& 0.60 0.70 0./0 0.

+i, -alculate the probabilit& that co!ponents chosen at ran"o! fro! the 11 co!ponents will both

function for a perio" of 2 hours fro! the co!!ence!ent of operations without breain' "own.

+ii, If at the en" of 2 hours of operations neither of the co!ponents chosen in+i, has broen "own$what is the probabilit& that the& are both t&pe - co!ponents. >12M?

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AAOC+C111(MATH+F113)