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Reg.

No.

B.Tech.

DEGREE

EXAMINATION,

NOVEMBER

2016

Third

Semester

15MA207

-

PROBABILITY AND QUEUING TI{EORY

(For the candidates admitted during the academic year 2015

-

2016 onwards)

(Statistical tables to be supplied) Notel

(i)

Part

-

A

should be answered in OMR sheet within

frst

45 minutes and OMR sheet should be handed over to hall invigilator at the end of 45m minute.

(iD

Part - B and

Part

- C should be answered in answer booklet.

Time:

Three Hours

Max.

Marks:

100

PART

-A(20x1=20

Marks)

Answer

ALL

Questions

(

-x,/

l.

A RV X

has the

probability

density

firnction

-f

(*)=lk' ",

*

>

0.

The

value of k

is

lo,

x<o

4'

Giu"n

E(X)

-1,

E(X\

=

2

for

a

discrete

RV X,

then

Var(3X+l)

is

(A)

2

(c)

I

Let

X

be a continuous

RV

then

(A)

F(-oo):

0

(C)

F(m;

-

-1

Let

X

be a

discrete

RV

with

possible respectively. Then

E(X)

is

(A)

1

(c)

3

(A)

e

(c)

27

(A)

z-

1

(C)

X:

P

(B)

(D)

(B)

F(oo)

:0

(D)

F(m)

-

2

values

1,2,3

associated

with

probabilities

""-'"

1,1.1

4'z'

4

(B)

2

(D)

4

(B)

r8

(D)

3

(B)

Z:6

(D)

x-

o

I

2

-l

2

1.

5.

If

X

is

uniformly

distributed in

(-3,

3) then its

probability

density

function

f(x)

is

given by

6.

The normal

probability

curve

is

symmetrical

about

(B)

1

4

(D)

1

9

(A)

1

6

(c)

1

3

(6)

Z.

If

the

RV X follows

a Poisson

distribution

with

mean

|

,f,"r.r

P(X:0)

is

given by

2

(A)

e-r

-1

(B)

e2

,

@)"2

^

(C)

ez

8.

tf th" MGF

of a

distribution

i"

"3(e'-1)

then the variance is

(A) 1

(B)

1

(D)

3

(c)

e

9.

The Chi-square test is used to

test

(A)

The

difference between population

(B)

The

difference between

population

means

variances

(C)

The

goodness

of

fit

(D)

The difference between

proportions

1

0.

If

a researcher rejects a

null

hypothesis

which

is true then

it

is a

(A)

Type

1

error

(B)

Type 2

error

(C)

Types

A

error

(D)

Standard

error

11.

The

't'

distribution

tends to

_

distribution

for sufficiently

large

value

of

y

(A)

Uniform

(B)

ExPonential

(C)

Standard

normal

(D)

Poisson

12.

_is

used

to

test the

difference

between the population variances

(A)

't'-test

(B)

Chi-Square test

(C)

Z-test

(D)

F-test

13.

The

symbol

c in

the

queueing

model

(alblc): (dle) stands

for

(A)

System

capacity

(B)

Number of

servers

(C)

Queue

size

(D)

Queue

discipline

14.

In

a queueing system the

arrival

and inter service rate respectively are denoted as

(A)

l,p

(B)

1,1

p'A

(c) I

(D)

,7

llt '

).p

4'-15.

If

the behavior

of the

queueing system does

not

depend on

time,

then the system is said

to

be

in

(A)

Transient

state

(B)

Idle

state

(C)

Steady

state

(D)

Busy

state

16.

If

)"

-8

/ hr

and

E(N)

-

4

customers

for

(MlMll):(mlFIFO)

queue system

then

p is given

by

(A)

e

(B)

10

(c)

1r

(D)

12

17. A Markov

chain

is

said to be

a-periodic

if

(A)

dr=l

(B)

di=Z

(c)

di:o

(D)

3

Page 2

of5

2INF315MA207

9

(7)

18.

Irthe

tpm

ora

Markov

"n"t"

t,

[fi

irf""or.,

=

(;,+)

then

ptrri,

1

19.

The reaction

to

state

i

is uncertain

if

Fii

:Ir,(')

it

(A)

[t lll

l-.-l

112

12)

(c)

[tt

t

I

la'o)

(A)

Atrnost

(C)

Exactly

(B)

[tt

t-

t:-]

-t

lzt'

z+

)

(D)

[r:

rrl

l'*'u)

j=1

(B)

0

(D)

Less than

I

(A)

Greater

than

1

(c)

1

20. A

Markov chain

is said

to

be absorbing

if

it

has one absorbing rate

@)

Atleast

(D)

Nearly

PART-B(5x4=20Marks)

Answer

AI{Y FIVE

Questions

21.

The

probability

distribution

of

X

is

Find

the mean and variance of

X.

Buses

arrive at

a

specified

stop

at

15

min

intervals

startirrg

atl

A.M,

that

is they

arrive

at 7,

7;15, 7:30,

7:45

and

so on.

If

a

passenger

arrives at the stop

at a

random

time

that

is

unifonnly

distributed

betu,een

7

and

7:30

A.M. Find

the

probability

that

he

waits atleast

12

min

for

a bus.

A

sample

of

size

13 gave an estimated

population

variance

of

3.0,

while

another sample

of

size

15

gave an

estimate

of

2.5-

Could both

samples

be

fiom

populations

with

the

same variance?

A

student's

study

habit's

are as

follows:

If

he studies one

nighthe

is 70%

sure

not

to

study

the

nexl

night.

On the other hand,

if

he does

not

study one

night,

he

is 60%

sure

not

to

study the next as

well.

Write

the tpm of the

Markov

Chain.

Explain

the

symbolic

representation

of

a Queuing model.

Given

the

RV X

with

density

tunction

f

(x)

=

{?.'

l '] '

I

n,ra the

pdf

of

Y:

8X3. 10, elsewhere

State and

prove

memoryless properfy

of

exponential

distribution.

22.

23.

24.

25.

26.

27.

x:

0 .,, 4 6

P(x):

I

6

I

;

J 8

I

:

3

E

(8)

PART-C(5x12=60Marks)

Answcr

ALL

Questions

28.

a.i. A

continuous

RV

X

has

pdf

(x):k

(l-x)

for

0 <

x <

l.

Find

the rth moment about

the

origin.

Hence

find

mean, variance.

ii.

If

X

denote the

number in

a

throw

of

a

fair die furd

E(X),

E(9x+2), Var

(X).

(oR)

b. A

fair

die

is tossed

600 times. Use TchebychefPs inequality

to

find

a

lower

bound

for

the

probability ofgetting

80

to

120 sixes.

29.

a.

Find

the

MGF

of

Binomial

distribution

and hence frnd the mean and variance.

b.

In

a

normal

distribution,

\Yo

of ther*::?"

under

35

and \9o/oare

under 63. What

are

the

mean and standard

deviation of

the

distribution?

30.

a. A

group

of

5 patients treated

with

medicine

A

weigh

42,39,48,60 and 41

kg,

a second

group

of

7 patients

from the

same

hospital

treated

with

medicine

B

weigh

38,42.56,64,68,69 and

62.Do

you

agree

with

the

claim

that medicine

B

increases the

weight significantly?

(oR)

b.

In

a

locality

100

persons

were

randomly selected and asked about

their

educational achievements. The results are

given

as

Education

Middle

High

school College

Total

Sex

Male l0

15

25

50

Female

25

10

15

50

Total 35

25

40

100

Can

you

say that education depends on sex?

31.

a.

Arrivals

at

a telephone

booth

are considered

to

be

Poisson

with

an

average

time

of

10

min

betw,een one

arrival

and the

next.

The length

of

a phone

call is

assumed

to be

distributed

exponentially with

mean 3

min

(i)

Find the average number

of

persons

waiting

in the system

(ii)

What is

the

probabiliff

that

a person

arriving

at

the booth

will

have

to wait in

the queue?

(iii)

What is

the

probability

that

it will

take

him

more

than

10

min

altogether to

wait for

phone and complete

his

call?

(iv)

The

telephone department

will

install a

second

booth when convilced that

an

arrival

has

to wait

on

the

average

for

atleast

3

min for

phone.

By

how

much

the

flow

of arrivals

should increase

in

order to

justify

a second booth?

(oR)

b.

In

a

single

server

queuing

system

with

Poisson

input

and exponential service times.

If

the mean

arrival

rate is

3

calling units

per hr, the mean service rate is

4

per

hr

and the

maximum

number

of

calling

units

in the system is 2,

find

(i)

P,(n>O)

(iD

Average

number of calling

units in the system

(iii)

Average

waiting

time in

the syslem

(iv)

Average

waiting

time in

the queue.
(9)

lo.z

0.3

32.u

TheQmof

aMarkov

chatn

{X^,n>0}havingthree

states

0,

I

and,

t,

"=

|

0.,

0.6

Lo.o

0.3 The

initial

distribution

is given

by ptol

=

(0.5

0.3 0.2) frnd

(i)

P(x,

-21

(iD

P(X3=3,Xr=2,Xr=l,Xs=2)-(oR)

b. A

salesman's

territory

consists

of 3 cities

A, B

and

C.

He never sells

in the

same

city

on successive days.

If

he sells

in

crty

A,

then the

next

day

he

sells

in

B.

However,

if

he

sells

either in

B

of

C,

then

the next

day he

is

twice

as

likely to

sell

in city

A

as

in

the

other

city.

How often

does he sell

in

each

ofthe

cities

in the steady state?

0.sl

031,

031

(10)
(11)

ii.

ln

a normal distoibution, 7Yo

of

the items are under 35 and 89% are under 63. What are the mean and standard deviation of the

distribution?

30.

a.i.

kr

a large

city

A,

20%

of

a random sample

of

900 school boys had a slight physical defect.

In

another

large

city

B,

18.5%

of

a

random

sample

of

1600

school boys had

the

same defect. Is the difference between the proportions significant?

ii'

The mean height and the SD

height

of

8

raadomly

chosen soldiers are 166.9 and 8.29 cms respectively. The corresponding values

of6

randomly

chosen sailors are 170.3 and 8.50 cm respectively. Based on these data. can we conclude that soldiers are, in general shorter than sailors?

b. A

total

number

of

3759

individual.

19.? ,n.*r"wed

in

a public opinion

survey

on

a

political

proposal.

Of

them, 1872 were men and

tle

rest women.

A

total

of

2ZS7

individuals were'in favour of

the proposal and 917 were opposed

to

it.

A

total

of243

men were undecided ar..d 424 women were opposed to the proposal.

Do

youjustiff

or contradict the hypothesis that there is no association between sex and attitude?

31.

a.

Customers

arrive at

a

one-man barber shop according

to

a Poisson process

with

a mean inter

arrival

time

of

12

min.

Customers spend an average

of l0 min

in the barber's chair.

(1)

What is the expected number of customers

in

the barber shop and in the queue?

(2)

Calculate the percentage

of time

an

arrival

can

walk

straight into the barber's chair

without

having to

wait.

(3)

How

much

time

can a customer expect

to

spend in the barber's shop?

(4)

What is the average

time

customers spend

in

the queue?

(5)

What is the

probability

that the

waiting time in

the system is greater than 30 min?

(6)

Calculate

the

percentage

of

customers

who

have

to wait prior to

getting

into

the barber's chair,

(oR)

b.

The

local

one-person barber shop can accommodate and

maximum

of

5

people

at

a time

(4 waiting and

1

getting

hair-cut).

Customers

arrive

according

to

a

Poisson

distribution

with

mean 15 per

hour.

The

barber cuts

hair

at an average rate

of4

per

hour

(exponential service

time)

(l)

What percentage of time is the barber idle?

(2)

What fraction

of

the potential customers are tumed away?

(3)

What is the expected number

of

customers

waiting for

a hair-cut?

(4)

How

much

time

can a customer expect

to

spend in the barber shop?

32.

a.

A

man either drives a car or catches a

train to

go

to

oflice

each day. He never goes

2

days

in

a

row by

train

but

if

he drives

one day, then

the next

day

he

is

just

as

likely to

drive again as he

is to

travel by

train.

Now

suppose

that on

the

first

day of

the week, the man tossed a

fair

dice and drove to

work

if

and

only

if

a 6 appeared.

Find

(i)

the

probability

that he takes a train on the

third

day

(ii)

the

probability

that he drives to

work

in the long run.

(oR)

b.

Three boys

A, B

and

c

are

tfuowing

a

ball

to

each

other.

A

always

throws

the

ball to

B

and

B

always

tfuows the ball

to

c,

but

c

is

just

as

likely

to

throw

the

ball

to

B

as

to

A.

Show that the process is

Mmkovian.

Find the transition

matrix

and classifu the states.

Reg.

No.

B.Tech.

DEGREE

EXAMINATION,

MAy

2012

Third /

Fourth Semester

I5MA2O7

_

PROBABILITY AND

QUEUING

TT{EORY (For the candidates admitted during the academic vear 2015

-

2016 onwards)

(Statistical table to be provided)

Part -

A

should be answered in OMR sheet within first 45 minutes and OMR sheet should be handed

over to hall invigilator at the end of 45m minute.

Part - B and Part - C should be answered in answer booklet. Note:

(i)

(ii)

Time:

Three Hours

].

A

random variable

X

has the

The value

of

K

is.

(A)

2/24

(c)

7/12

2.

var[+x

+

g]

is

(A)

rzvar[xj

(c)

t6var[x]

PART-A(20x1=20Marks)

Answer

ALL

Questions

Max.

Marks:

100

3.

When a die is

t}rown, X

denotes the number that tums up,

find

the mean

@)

2U24

@)

1/24

(B)

avar[x]+s

(D)

16

var[x]+

s

@)

3/4

(D)

-<ls

(B)

125, 3.674

(D)

12s,2.674

l-

pe'

(A)

7t2

(c)

4/s

4.

Given that the PDF

of

a random variable

X

is

/(x)=Zx; 0<x<l,find

p(,Y>0.5)

(B)

2t3

(D)

4/s

(A)

1/2

(c)

3/4

5.

If

on an averagg 9 ships out

of

10

arrive

safely

to

a

por!

then mean and

sD

of the number

of

ships

returning

safely out

of

150 ships are

(A)

13s,2.674

(c)

13s,3.674

Mean of the Poisson

distribution

is

(A)

l.

(c)

1/tu

7.

The

MGF of

Geometric

distribution

is

(A) 1

(B)

l-q"'

(B)

?,+1

(D)

T,

(D)

J!-l-q"'

function

0

I

2 3 4

P(x): k 2k 5k 7k 9k

PaEe 4 of 4

22MF3l4l5Il,A207

page

I of4

(c)

L-

pe
(12)

8.

The variance of

Uniform

distribution

U(a,b)

is

20. A

state

'i'

is said to be aperiodic

with

period

'di'

if

(A)

I

(,r

-

r)'

12'

'

(c)

1(a*o)

2'

9.

The value set

for

cr is

known

as

(A)

Rejection level

(C)

Signihcance level

(B)

(D)

Lr@-")'

)o-,)

(B)

Acceptance

level

(D)

Hypotheti cal

level

(B)

Standard

enor

@)

Sample

enor

@)

Population, sample

(D)

Parameter, statistic

(B)

Alternatehypothesis

(D)

Optionalhypothesis

Uniform

Exponential

(B)

rP=n

(D)

nP

=r

(B)

Finite

(D)

Full

(A)

di<1

(c)

di>1

(B)

di:

I

(D)

di:

o

PART-B(5x4=20Marks)

Answer

ANY FfVE

Questions

10.

The standard deviation

ofa

sampling

distribution

is called as

A

continuous random variable

x

has a

pdf

/(x)

=

laze-';

x>

0 ' Find the value

of

k'

Given the random variable x

with

density

function

l2x

o<x<1

/(x)

=.{

-"

. Find the pdf of

Y

:

8X3.

r

\ '' L0

elsewhere

If

the

probability

that an applicant

for

a

driver's

license

will

pass the road test on arry.given

trial

iJ 0.8, what is the

probability

that he

will

finally

pass the

test (i)

on the

fourth

hial

(ii)

in

fewer than 4

trials.

The

mileage

which

can

owner

get

with

a

certain

kind

of

radial

tire is

a random variable having an exponential

distribution with

mean 40, 000 km. Find the probabilities that one

of

these tires

will

last

(i)

atleast 20,000kms

(ii)

atrnost 30, 000kms.

Define

(i)

null

hypothesis

(ii)

altemate hypothesis

(iii)

Bpe

I

error

(iv)

type

II

error.

what

do

the letters

in the

symbolic

representalion (a

/b

I

c):(d

I

e)

of

a

queuing model represent?

27.

Ifthepmof

aMarkovcirainis

[,:"

.

]".l

Findthesteadystatedistributionof

thechain.

11/2

1t2)

2t.

(A)

Sampling error

(C)

Simple error

11.

A

is a subset

of

a

23.

24.

12.

The hypothesis that an

aralyst

is

trying to

prove is called

(A)

Sample, population

(C)

Statistic,parameter

(A)

Electivehypothesis

(C)

Null

hypothesis

(A)

aP=o

(c)

z(t-r)=o

(A)

Empty

(C)

Ergodic

13.

The symbolic notation

of

queuing

model

is represented

by

(A)

Kendall

(B)

Euler

.

(C)

Fisher

(D)

Neuman

14. The interval

between

two

consecutive arrivals

of

a

Poisson

process follows

distribution.

(A)

Binomial

(C)

Normal

(B)

(D)

(A)

Balking

(C)

Jockeying

(B)

Reneging

(D)

Leaving

(B)

(D)

25.

26.

28. a.

ll.

29. a.

15.

The

probability

of

'n'

customers in the system Po

:

,o,

[4)r.

\p)

'"'

(#)'",

16.

Which

term refers to

"a

customer

who

leaves the queue because the queue is

too long"

l)

,0

Po

(

s"\

tt

lp)

(+)

PART-C(5x12:60Marks)

Answer

ALL

Questions

A

random variable

X

has the

following probabilit

x:

|\

-1

0

I

2 3

P(x): 0.1

k

0.2

2k

0.3

3k

distribution

(i)

Find

k (ii)

evaluate

P(X

< 2)

(ii|

P(-2

<

X

<

2)

(iv)

CDF of

X

(vi)

variance of

X.

(v)

mean

of X

(oR)

If

X

is a

random variable

with

E(X)=3

and

EOp):13,

find

the

lower bound for

P(-2

<

X<

8 ), using

Tchebycheff

s inequality.

If

X

represents the outcome,

when

a

fair

die

is

tossed,

furd

the

MGF

of

X

and hence

find

E(X)

and var

(X).

out of

800 families

with

4 children each, how many families

wouid

be expected to have

(i)

2

boys

and

2 girls

(ii)

atleast

I

boy (iii)

atmost

2 girls

(iv)

children

of both

sexes. Assume equal

probabilities for

boys and girls.

(oR)

Buses arrive at a specified stop

at

15 mins

interval

starting at 6 am

(ie)

arrive at 6am' 6.15

am, 6.30

am and

so

on.

If

a

passenger

arrives at

the

stop

at a time that is uniformly

distributed

between

6

and 6.30 am,

find

the

probability

that he

waits

(l)

less than

5

mins

for

a bus (2) more

than

10 mins

for

a

but.

,2MF3/4r5r\ra207

17.

The sum of

all

the elements

of

any

row

of

the hansition

probability matrix

is

(A)

0

(c)

0.7s

(B)

0.s

(D)

I

The

steady state

probability

vector

r

of

a discrete

markov

chain

with

transition probability

matrix

satisfies the

matrix

equation.

18.

19. A

non-null persistent and aperiodic state is called

b.i.

(13)
(14)
(15)
(16)
(17)
(18)
(19)

on

Ior

lne

Iollo

0

.r 1 2 3 4 5 6 Total

f

5 18 28

t2

-, 6 4 80

ii.

A

continuous random variable

X

has a

pdf

f

(x)

=

b2

"-*

,

frnd

h

mean and variance.

29. a.

Fit

a

binomial distribution

for

the

foll

data:

30.a.

(oR)

In

an engineering

examinatioq

a student

is

considered

to

have

failed,

secured second class,

fust

class

and

distinction

according

as

he

scores less

than

45olo,

between

45Yo artd 600/o,

between 60% and 7 5%o and above 7 5o/o respectively.

In

a

particular year

l0o/o of the students

failed

in

the

examination and

5o/o of.

the

students

got

distinction.

Find the

percentages

of

students

who

have got

frst

class and second class.

15.5 percent

of

a random sample

of

1600 undergraduates

were

smokers, whereas 20Vo

of

a

random

sample

of

900 post-graduates

were

smokers

in

a

state. Can

we

conclude

that

less

number

of

undergraduates are smokers than the postgraduates?

(oR)

Theory predicts that the proportion of beans in four groups

A, B,

C and

D

should be 9:3:3:1. In an

experiment among 1600 beans the numbers

in

the four groups were 882, 313, 287 arrd 118. Does

the experiment support the theory?

Arrivals at a

telephone

booth

are

considered

to

be

Poisson

with

an

average

time

of

12

minutes between one

arrival

and

the next. The length

of

a

phone

call is

assumed

to

be distributed exponentially

with

mean 4 minutes.

(i)

Find the average number of persons

waiting

in the system.

(ii)

What

is the

probability

that a person

arriving

at

the

booth

will

have

to

wait

in

the queue?

(iii)

Estimate the

fraction

of the day when the phone

will

be in use'

(iv)

What

is

the

probability

that

it

will

take

hirn

more

than

10

minutes

altogether to

wait for

the phone and complete his call?

31. a.

Reg. No.

B.Tech.

DEGREE

EXAMINATION, MAY

2018 First to Sixth Semester

t5MA207

-

PROBABILITY AND

QUEUING TFIEORY

(For the candidates adnitted during the academic year 2015

-

2016 omtords) Note:

(i)

Part -

A

should be answered

in

OMR sheet

within first

45 minutes and OMR sheet should be

handed over to hall invigilator at the end of 45d minute.

(iD

Part - B and

Part

-

C

should be answered in answer booklet.

Time: Three Hours Max. Marks: 100

PART-A(20x1=20Marks)

Answer

ALL

Questions b.

1.

If

C is a constant then

E(c)

is

(A)

0

(c)

cf(9)

2.

Var(4X

+8)

is

(A)

l2Var(X)

(C)

16Var(X)

3. A

random

variable

X

has

mean

p

:

distribution, then

P(6

<,Y

< 18) is

(A)

I

2

(c)

I

4

(A)

lt.-

np,

o2

= npg

(C)

H=ne,o2

=npe

6.

Mean of the Poisson

distribution

is

(A)

r"

(C)

I'

7.

The

MGF of

geometrical

distribution

is

(A)

1

l-

q"'

(c)

q

l-

pr'

Page 1 of 4

(B)

4Var(X)+8

(D)

l6Var(X)+8

12

and variance

d:9

and

(B)

p

=npq,oz

=np

(D)

p

-np,

o2

=

pe

(B) l.+

t

(D)

x-l

(B)

1

l-

pu'

@)

pet

7-

qu'

an unknown probability

(B)

I

(D)

c b.

(B)

1

4

@)1

8

(oR)

b.

The local

one-person

barber

shop can

accommodate

a maximum

of 5

people

at

a

time

(4

waiting

and

lgetting

hair-cut).

Customers

arrive

according

to

a Poisson

distribution

with

mean 5 per hour. The barber cuts hair at an average rate

of4

per hour.

(i)

What percentage

of time

is the barber idle?

(ii)

What

fraction

of the potential customers are tumed away?

(iir)

What is the expected number

of

customers

waiting

for

a hair-cut?

(iv)

How

much

time

can a customer expect

to

spend in the barber shop?

32.

a. A

gambler has

t2

he bets

{1

at a

time

and

wins

{1

with

probabiliLy

112.

He

stops

playing

if

he loses T2 or wins

{4.

(i)

What is the

tpm of

the related

Markov

chain?

(ii)

What is the

probability

that he has lost his money at the end

of

5 plays?

(iiD

What is the

probabiliff

that the game lasts more than 7 plays?

(oR)

b.

Three boys

A,

B

and C are

throwing

a ball

to

each other.

A

always throws the ball to

B

and

B

always throws the ball

to

C, but C is

just

as

likety

to

throw

the ball

to

B

as to

A.

Show that the process in

Markovian.

Find the transition

matrix

and

classiff

the states.

:F****

4'

rc

n(xz)

-8

and

E(X)

-2,then

Var(X)

is

5.

The mean and variance

of

a

binomial distribution

is

(B)

2

(D)

4

(A)

3

(c)

1
(20)

19.

If

one step transition

probabiliry

does not depend on the step, then the

Markov

chain is called

8.

If X

is

uniformly disfibuted

in

(0,

10), then

P(X >

8) is

(A)

Reducible

(C)

Homogeneous

20.

The

steady state

probability

vector

matrix

equation

(A)

nP:0

(C) nP:x

(B)

Regular

(D)

Non homogeneous

z

of

A

discrete

Markov

chain

with

TPM

satisfies the

(B) zP:p

(D)

n(l+P)

-

0

2,3and4such

probability

distribution

(A)

r/s

(c)

3/5

9.

The

form

of the altemative hypothesis can be

(A)

One-tailed

(C)

Neither

one nor

two

tailed

(A)

Sampling

enor

(C)

Standard error

10.

What is the standard deviation

of

a sampling

distribution

called?

(B)

r/ro

(D)

1/3

@)

Two-tailed

@)

One or

two tailed

(B)

Sample error

(D)

Simple error

12.

Student's

t-distribution

has

(n-1)

sample are

(A)

Dependent

(C)

Maximum

13.

What stand

for

'd'

in the queue

model

(a I b I

c:d

/ e)

(A)

Queue discipline

(C)

Service

time

degrees

of

freedom. When

all

the

n observations

in

the

(B)

Independent

(D)

Minimum

ffi,-.3,$;fl;1ffim

If X

is uniformly distributed

t f+,+)

find the pdf

of

.l'

=tanX

\2'2

)

If

the

probability

that an applicant

for

a

driver's

license

will

pass the road test on any

given

trial is

0.8, what is the

probability

that he

will

finally

pass the test

(i)

on the

fourth

trial

and

(ii)

in

fewer than 4

trials?

The

fataliq,

rate

of typhoid

patients

is believed

to be

l7.26percenl In

a

certain yeat

640

patients suffering

from

typhoid were

treated

in

metropolitan hospital and

only

63

patients died. Can you consider the hospital efficient?

In

the usual notation

ofa(M

lM

/l):(a/

FIFO)

queue system

ifX:

12

perhour

andp:24

per hour,

find

the average number

of

customers in the system and in the queue.

The

transition

probability

matrix

of

a

Markov

chain

{X,

} , n

=1,2,3....

having 3

states 1, 2, (

o.t

o.s

0.4)

tt

and

3

is

P=l

tt

0.6

0.2

0.2

I

and

the

initial

distribulion

is

p(0)=(0.7,0.2,0.1).

Find

[0.3 0.4

0.3

)

P{x2

-3}.

21. I

1.

When the researcher rejects a true

null

hypothesis, a error occurs.

(A)

Type

I

(C)

Type

B

@)

TypeA

(D)

Type

tr

z3-24.

25.

t4.

p-1"

(c)

x

Lr+I

(A)

Balking

(C)

Jockeying

(A)

0.18

(c)

0.38

(A)

Present

(C)

Future

18.

Ergodic means

(A)

Ineducible

and

periodic

(C)

Not

ineducible

(B)

System capacity

(D)

Number

of

servers

l"-p

(D)

p

r+p

(B)

Reneging

(D)

Leaving

(B)

Past

(D)

Outcome

(B)

kreducible

and aperiodic

(D)

Regular

The average number

of

customers

in

the system

n

(M

I

M

/

l:

a

I

FIFO)

model is

(A) l,

(B)

p

15.

Which

term

refers

to

oa customer.who leaves the queue because the queue is

too long'?

If X

is

a random

variable

whose density

function

i,

"f(x)

= €--x ,-0 <

r

< co.

Find

rth

moment

-

0 otherwise about

origin

andhence

find

p1

and

12.

State and prove memoryless property of the exponential

distribution.

16. In (M

I

M

/l:K

/FIFO)

model,

if

)":3

perhour,

pt:4

perhour

and

effective

mean

arrival

rate

of

a customer is 2.88 per hour then what Pn?

26.

a1

(B)

0.28

(D)

0.48

17.

Markov

proves is one

in

which the future value is independent

of

values.

PART-C(5x12=60Marks)

Answer

ALL

Questions

28.a.

If

the

random

variable

X

takes

the

values

1,

2P(X

-l)=3P(X

-2)=

P(X

=3)

-sP(X

-4),

find

the cumulative

distribution function of

X.

that

and

(oR)

b.i

A

fair

die is

tossed

720

times.

Use

TchebychefPs

inequality

to find

a lower

bound

for

the

probability

of getting 100

to

140 sixes.
(21)
(22)
(23)

ii.

Derive the moment generating fimction of

uriform

dishibution and hence firrd mean and variance

ofthe

distribution.

(oR)

b.i.

If

X

has

a

geometdc distribution,

then

for

any

two

positive

integers

m

and

q

prove

that

PIX>m+n/X>m]=PlX>nl.

ii.

Let

X

be

is

a

normal

variate

with

mean

30

and standard deviation

is 5. Find

ttre

following

Pl26

<

X

<

401,

plx

>

4sl

and

P1:lx

4oll>

5.

30.

a-

Two random samples gave the data.

SamDle Size Mean Variance

I

2

8

li

9.616.5

t)

2.t

Cao we conclude that the two samples have been drawn from the same normal population?

(oR)

b.i.

Test the significance

ofthe

difference between the means

ofthe

samples, dm\,,,rr from two normal populations with the same standarddeviation

SamDle Size Mean SD

I

2 100 200 61 63 4 6

ii.

The following data give the number

ofair-craft

accidents that occured during the various days

of

a

week-Days

Mon

Tues Wed

Tturs Fri

Sal

No. ofaccidents

l5

19

l3

t2

t6

15

Test whether the accidents are

unifomly

distribuGd.

31.

a.

A

deparlmental store has a single cashier. During the rush hours, customers araive at the rate of 20 customers per hour. The average number

of

customers that can be processed by the cashier is 24 per hour.

(i)

what

is

the probability that the cashier is idle?

(ii)

what is the average

number

of

customer in the queuing system?

(iii)

what is the aveEge time a customer spends in the system?

(iv)

whal is the

avemge number

of

customers

in

the

queue?

(v)

what

is the

average

time

a customer spends in ihe queue, waiting for service?

(oR)

b.

The local one-person barber shop can accommodate a maximum

of

5 people at a time (4 waiting and

I

getting hair cut). Customers ardve according to a Poisson distribution with mean 5 per hour.

The

baxber cuts

hair at an

avenlge

mte

of

4

per hour

(exponential service

time).

(i)

what

percentage of time is the barber idle? (ii) what ftaction

ofthe

potential customers are tumed away?

(iii)

what is the expected nunber

of

customers waiting

for

a hair

cufl

(iv)

how much time can a customer expected to spend in tl-re barber shop?

32.

a"

The transition probability matrix

of

a Markov chain

{\}:

n

= t,

2,

3...

having 3 states 1, 2, and 3

fo.r

o.s

0.4\

tt

is

-

P=l

tt

0.6

0.2

0.21.

Initial

distribution

is

P(0)

=

(0.7 0.2 0.1).

Find

(D

P{Xr=3}

(0.3 0.4

0.3,

(ii) P{x3=2,

&=3, xr=3,

&:2}

(oR)

b.

Three boys

A,

B

and

C

are throwing a ball

to

each other.

A

always thiows the

ball

to

B

and

B

always tl.[ows the

ball to

C, but C is

just

as

likety

to

tlrow

the

ball

to

B

as to

A.

Show

tlEt

the

process is Markovian. Find the tmosition*matrix and classi8, the states.

Reg. No.

B.Tech. DEGREE

EXAMINATION,

NOVEI\{BER

2015 Fourth Semester

.

MAIO14-

PROBABILITY AND

QI]ET]ING

TMORY

(For the candidates adnixed

d

ring ,he academic ))ear 2013

-

2014)

(Statistical tunb is to be provided)

Part - A should be a.nswered in OMR sheet within first 45 minutes ard OMR sheet should be ha.nded over to Note:

(,

hall iflvigilator ar the end of456 minute.

(i,

Part - B ard Part - C should be answered ifl arswer bookleL

Time: Three Hours

1.

Max. Marks: 100

PART

-A

(20

x 1=

20

Marks)

Answer

ALL

Questions

The probabiliry density

imction

ofa

contiruous random variable

is

f(x\:ce-'l;-a<x<a

then the va1u9

ofc

is

(A)

v2

(c)

3/4 dala.

z

altt

lf

"

/(r)=i

'

l0

(A)

0.33

:-

"-'-,

then

E(x)

is

(B)

1/4

(D)

1/6

(B)

(D)

(c)

2.33

3.

Yd44X+8) is

(A)

t2va{x)

(c)

t6ra4x)

1.33

(B)

4,za{x)+8

@)

161/a(x)+8

(B)

P

factorial moment

(D)

(r-l)d

cenaal moment

@)

Poisson

(D)

Normal

(A)

rd centlal moment

(C)

rd mw moment

1tt

(121=le,'1,,

-*.

r,.*

I

2Ja

5.

Which distribution has equal mean and variance?

(A)

Binomial

(C)

E).?onential

(A)

e-t

l;

x>o

(c) l-e-b;

x>o

4

IfX

is a random variable and r is an integer, then

E(-t

-t)'

represents

6.

Ifthe

probability that a target is destroyed on any one shot is 0.5, then the probability that it

would

be destroyed on 6d attempt is

(B)

0.0165

(D)

0.0156

(A)

0.106s

(c)

0.r056

7.

The cumulative distdbution fimction

off(x)

ofan

exponential distribution is

@)

"t'

-t;

x>o

@)

1-etu;

.r > o

8.

The prcbability density function of a standard normal distribution is

1

2l^

(ts)

0e\=-=e

'

t'.0<z<*

\l

zlt

1

)/^

(D)

0lzl=-=e' t'i-a

< z

<a

\lZJr

rct

drzt

=Le'zl,g.

r.*

"l2z

(24)

10.

The test used to test the equaliry ofvariance

oftle

populations &om two small samples is

(B)

F-test

(D)

Independent of altributes

20.

State

i

is said to be aperiodic with period

di,

if

(A)

di>

1

(B)

d'<1

(c)

4=1

(D) di=o

(A)

t-test

(C)

t'

.test

s/ ',J n

X-"

ll:l

-

7=

' sf

.ln

-r

In large samples, to test the sigrfficance

ofthe

difference between sample mean mean p, the test statistic (sample standard deviation s is known) is

al

,=P,

!

At

,=-!J:

slJn-l

.{

and population

21NA4nA10t,l

&=gardan

0<x<l

elsewhere

ulnown

probability

the

steady state

PART-B(5x4=20Marks)

Answer

ANY FIVE

Questio.s

A

random variable

X

has

a

mean

p

=

12 and variance distribulion. Find P(6<X<l 8).

22.

23.

25.

26. 21.

Given the random variable

X

with density fimction Find the

pobability

densiry

ofY

= 8X3.

A

favel

company has

two

cars

for hiring.

The demand

for

a cztl on each day

is

distributed as

Poisson variate with mean 1.5. Calculate the

propotion

ofdays on which some demand is refused'

24,

The

time (in hous) requied to

repair

a

maohine

is

exponentially distributed

with

parameter

L

=

U2. What is

the conditional

probability

that

a

repair takes a.tleast 10

hous

given

that

its duration exceeds

t

hou$?

Experience has

shoun thal

20'/o

of a

manufactured product

is

of

top

quality.

In

one

day's production

of

400 articles

only

50 are

of

top quality. Show that either the production

of

the day

chosen was not a reprcsentative sample or the hypothesis of 20% was wrong.

In

a railway marshalling yaxd goods trains aEive at rate

of

30 trains per day. Assuming that the inter-arrivai time

followi

an exponential distribution and the service time is also exponential

with

an average of 36 minutes. Calculate expected queue size (line length).

27.

If

the

ran:ition

probability marrir

of

a

Marko\

an", ,,

[,f,

distribution of the chain.

l2x

rr,)

=

to

I

l.

95% conidence limits for the population proportions is

(A)

p+2.s8P

rc)

1.96,@

tn

@)

,-,rrp

@

orr.rup

12.

The application of one-tailed or two-tailed test depends upon the nature

ofthe

(A)

Null

hypothesis

(B)

Altemative hypothesis

(C)

Level

ofsignificanoe

(LOS)

@)

Degrees

offieedom

13. In

queuing system, the mrmbet

of

customers sewiced per

unit

has a Poisson distribution

with

mean

(A)

p

(c)

2tv

In the symbolic representation

ofa

queue model (a I b /

c):(d

/ e), wlrere c denotes.

(A)

The type

ofdishibution

ofthe no

of

(B)

The type

ofdishibution oflhe

no ofservices

t

).

o,o

v2)

14.

15.

anivals per unit time

(C)

No ofservers

(A)

p,

>0

&l

p;i

=l

J

(c)

pii>O

&lpii

=1

(A)

Reducible

(C)

Periodic

(B)

l/p

(D)

3/p

per rmit time

(D)

The capacity

ofthe

system

(B)

Two-step transitionprobability

(D)

n-step transition probability

@)

ri1<0

&l

Pii

=l

@)

eii

<0

&f

P,t

=1

(B)

Ineducible

@)

Aperiodic

Arrivals

at a telephone booth are considered to the Poisson

with

an average

time

of

12 minutes between

one arrival

and

the

next.

The length

of

a

phone

call is

assumed

to be

distibuted

exponentially with mean 4 minutes. What is the average number of persons waiting in the system?

(A)

0.5

(B)

1.0

(c)

r.5

(D)

2.0

16.

Probabitity

tlat

the nurnber ofcustomers in the system exceeds k (i.e) P1N > k) is

(A)

)"lp

@)

(ilr)o

oo

(c)

@l

p)o*'

po

{o)

1tl p)k-t

eo

17.

The

conditional

probability

that the

4

at step 0 is called

(A)

One-steptrarsitionprcbability

(C)

(n-

1) step tra.sition probability

process

is

in

state

aj at

step

q

given

that

it

was

in

state

18.

The P',r of a Markov chain is stochastic matrix, because

PART-C(5x12=60Marks)

Answer

ALL

Questions 28.

a.i.

A mndom vadable X has the

fo

(A)

Find

K

(B)

Evaluate

P(X<

2)

(C)

Evaluate

P(-2<X4)

@)

Find the

cumulative distribution fimction

ofx

(E) Evaluate the mean and variance

ofx.

. Ihe

probability

tunction

b.r.

Ptx

=

i)

=

+lj

=1,2,....@). 2r

ofthe

disaibution.

I

-21.

ii.

rf

f(

=|xe

"

/':

r>u

[0

I

x<0

(l)

Show that

f(r)

is a Eobabiliry

deNity

function of a continuous random variable

X

(2)

Find its distribution F(.t).

19.

tf

pl"!o

for

some

n

and

for all

i

and

j

then every state can be rcached

fiom

every other state.

When this condition is satisfied, the Markov Chain is said to

(oR)

of an

infinite

dis$ete

distribution

is

given

by Verify that the total probability

is

1 and find the mean and variance

21NA4MAI014

(25)
(26)
(27)

Reg.

No.

B.Tech.

DEGREE

EXAMINATION, MAY

2018

Fourth

Semester

MA1014

-

PROBABTLITY AND

QUEITING

TIIEORY

(For the candidates admitted during the academic yeor 2013

-

2A14 and 2014 -2015) (Statistical tables to be supplied)

Note:

(i)

Part -

A

should be answered in OMR sheet within

frst

45 minutes and OMR sheet should be handed over to hall invigilator at the end of 45tr minute.

(ii)

Part - B and Part - C should be answered in answer booklet.

Time:

Three Hours

Max.

Marks:

100

PART-A(20x1=20Marks)

Answer

ALL

Questions

1.

The

probability density'

function

of a

continuous

random variable

is

f

(*)

-

C"-l*l

,-

co <

r

<

then the value

of

C

:

(A)

1

2

(c)

1

4

!=4x+3

is

(A)

]r'-tl

(c)

)O_rl

(A)

2

2-t

(c)

ze_t)-3

4'

6

random

variable

X

has

mean

F:12

distribution, then

P(6.

x

<

l8)

is

(A)

1

2

(c)

1

4

(A)

3

(c)

I

(B)

(D)

(B)

3

3-t

(D)

\z_t)-2

and

varian"".o2

=9

and an unknown

probability

1

4

T

6

2.

If

the

continuous

random

variable

X

has

pdf

/(x)

={:*'o<x<l.

rhen

the

pdf of

' [0,

otherwise

(B)

frfr_rl

(D)

)tr_rl

3.

A

random

variable

X

has the

pdf

given

by

f

(*\={'"-'*,'=',

then

MGF

is

Lo,

x<o

(B)

I

4

(D)

1

8

5'

If

X

is

a Poisson variate such

that

,(r')

=

o ,t.r-,

r(x)

is

(B)

2

(D)

0
(28)

6.

The mean of the

exponential distribution

with

pdf

le-L*,

x

)

0

(A)

)"

(c)

1,

"2 L

7.

Arandom

variable

X

has

4 r',niform distribution over

(-:,:).The

value

of

k

for

which

P( \,ra

x>r) =1

i,

-f

(B)

1

).

(D)

1

(B)

2

(D)

-2

(A)

3

(c)

1

8.

Normal distribution

is the

limiting form

of

conditions.

(A)

Binomial

(C)

Geometric

distribution

under suitable statistical

9. The Chi-square goodness

of

fit

test can be used to test

for

(A)

Significanci of

sample

statistics

(B)

Difference

between

population

means

(C)

Normality

(D)

ProbabilitY

Which

of

the

following

values is

not

typically

used

for

o?

(B)

0.0s

(D)

0.2s

(B)

Poisson

(D)

Uniform

(B) n-2

(D)

n-l

(B)

Type

A

(D)

Type

B

(B) 71

--r

u

(D)

)"

,

-+r

u

10.

(A)

0.01

(c)

1.10

1

1.

The degrees of freedom

for

t test based on

n

observations is

(A)

Zn-r

(c)

z(n-r)

12.

When the researcher rejects a true

null

hypothesis a

error

occurs?

(A)

Type

I

(C)

Type

II

13.

The

probabiliry

of

no

customers

in the

system

i"

(M

I

M

ll;a

I

FIFO)

model

(A)

L

u

(c)

,

.l

r

--u

14.

The

probability

that

the

number

of

customers

in

the

system

exceeds

lq

1n

(U

tU l!:olFIFO)

model

(A)

(

t

\o*t

lr)

(c)

(

^\o*,

ln)

(B)

(

t\o*'

lr)

(D)

(

t\o

ln)

(29)

15.

16.

What is the mean of the Poisson process?

(A)

I

(c)

)"tt

The

traffic

intensity of

a queuing system is

(A)

,2'

(C)

lt

P

(B)

^t

(D)

tt)"

(B)

p

(D)

ptl

17.

A

state

i

is said

to

be aperiodic

with

periodic di

if

(A)

di

<t

(c)

di

>r

(A)

Finite

(C)

Empty

(A)

Ep

=0

(c)

"(t-

p)=o

(B)

di

=1

(D)

di=0

(B)

Finite

(D)

1

(B)

Infrnite

(D)

1

(B)

nP=n

(D)

frP'=0

18.

A

non

null

persistent and

aperiodic

state is called

(A)

Eorpty

(C)

Ergodic

19.

The state

i

is

said to be non

null

persistent

if

its mean recurrence

time

pii

is

20.

The

steady state

probability

vector

z

of

a

discrete

Markov

chain

with

transition probability

matrix p

satisfies the

matrix

equation

2t.

22.

PART-B(5x4=20Marks)

Answer

All-Y

FM Questions

?

If

a random

variable

X

has moment generating

function

ltr*lt)

=

*.

Obtain the

standard

deviation of

X.

Trains arrive

at a station

at

15 minutes

intervals starting at

4am.

If

a passenger

arive

to

the statin at a

time that

is

uniformly

distributed

between 9.00 and 9.30

find

the

probability

that

he has to

wait

for

the

train for (i)

less than 6

minutes

(ii)

more

than

10 minutes.

Theory predicts that the

proportion of

beans

in four

groups

A, B,

C,

D

should be 9:3:3:1.

In

an

experiment

among 1600 beans,

the

numbers

in

the four

groups

were 882,

313,287

and

1i8.

Does

the

experiment support the theory?

Explain

Kendall's

notation for

queuing models.

State and

prove

memoryless

properly of

geometric

distribution.

Find

MGF

of

the random variable whose

moments

are

pr':

(r

+

1)3'

and

find

its mean. 24.

(tt+

1/4\

25.

Let

p=l-

._

_

-

l

be

the

TPM

of

a

two

state

Markov

chain.

Find

the

stationary

' lU2

U2)

probabilities of

the chain. 26.

27.

(30)

PART

-

C (5

x

12 -- 60

Marks)

Answer

ALL

Questions 28.

a.i

If

X

is random

variable

having the density

function

^,

\

fxl6,x=1,2,3

,r(r)

=

{0,

otherwise

Find

E(f

\

+2x+Z)

/

anO

var(4x+5).

\

"

(8Marks)

ii.

If

the

continuous

random variable

X

has

pdf

f

(x)=2/9(x+t),-t

<x<2

find

the

pdf

of

2

!

=

x".

(4 Marks)

(oR)

b.i

Th"

distribution

function

of

random

variable

X

is

given

by

lr(x):t-(1

+x)e-*,x>0.

Find

the

density

function,

meane variance

of

X.

(8 Marks)

ii.

A

fair die

is

tossed 720

times. Use Tchebycheff

inequality to

find

a

lower

bound

for

the

probability

of

getting

100

to

140

sixes.

(4 Marks)

29.

a.i

Fit

a Poisson

distribution for

on

lor

the

foll

data

x

0 I 2 J 4 5

Total

f

t42

156 69 27 5 1 400

Find the

probability

mass

function

and then

find

the

theoretical

frequencies.

(8 Marks) The

time (in

hours)

required

to

repair

a machine

is exponential distributed

with

parameter )"

=l

I

2.

What

is the

conditional

probability

that a

repair

takes

atleast

10h

given that

its

duration exceeds

th?

(4 Marks)

(oR)

b.i

In

an engineering

examination,

a student

is

considered

to

have

failed,

secured second class,

frst

class

and

distinction,

according

as

he

scores

less

that

45yo,

between

45%

and 60%

between 60%

and 75Yo and above

75o/o

respectively.

In

a particular

year

10%

of

the

students

failed

in

the

examination

and

5%

of

the

students

got

distinction.

Find

the percentages

of

students

who

have got

first

class and second class. (8 Marks)

ii.

If

the

probability

that

an applicant

for

a

driving

license

will

pass

the

road test

on

any

given

trial

is

0.8. What is the

probability

that he

will

finally

pass the test on the

fourth trial?

(4 Marks)

30.

a.i

Random samples

of

400 men and 600

women

were asked whether

they

would like to

have a school near

their

residence 200

men

and 325 women

were

in favour

of

the

proposal.

Test the hypothesis that

the proportions

of

men

and

women

in

favour

of

the

proposal

are same,

at5%

level of

significance. (8 Marks)

ii.

A

group

of five

patients treated

with

the medicine

'A'

weights

42,

39,

48,

60

and

4l

kg;

u second

group

of

7 patients

from

the

same

hospital treated

with the medicine

'B'

weights

38,

42, 56,64,68,

69

and 62

kg. Do you

agree

with

the claim

that

medicine

B

increases the

weight significantly?

(4 Marks)

(oR)

(31)

b.i

The

mean

of

2large

samples 1000 and

2000

members

are 67.5

inches

and 68

inches

respectively.

Can

the

samples be regarded as

drawn

from the

same

population

of

standard

deviation:2.5

inches? (8 Marks)

ii.

Two

independent samples

of

sizes

9

and

7 from

a

normal

population had

the following

values

of

the variables

Do

the estimates of the

population

variance

differ significantly

at 5Yo level? (4 Marks)

31.

a.

Customers

arrive

at a watch

repair

shop according

to

a Poisson process at a rate

of

one per

every

10 minutes and service

time

is an exponential random variable

with

mean 8 minutes.

(i)

Find

the average number

of

customers

Ls

in

the shop.

(iD

Find the

ayerage

time

a customer spend

in

the shop Ws.

(iii)

Find

the average

time

that a customers spends

in the

queue Wq.

(iv)

Find the

average number

of

customers

in

the queue

Lq.

(v)

What

is the

probability

that the server

is

idle?

(oR)

b.

In

a car wash service

facility,

cars

a:rive

for

service according

to

Poisson

distribution

with

mean 5 per

hour. The

time for

washing

and

cleaning

each car has

exponential distribution

with

mean 10 minutes

per

cax.

The.facility

cannot handle

more

than

one

car at a

time

and has

atotal of

5

parking

spaces.

(i)

Find

the

effective arrival

rate.

(ii)

What is the probability

that an

arriving

car

will

get

service immediately

upon

wrival?

(iii)

Find

the expected number

of parking

spaces occupied.

32.

a.

The

transition

probability

matrix

of

a

Markov chain

\Xr\,n--I,2,3...

having

3 states 1, 2

(r.,

o.s

o

o)

and

3

is

r=lo.o

0.2

0.2

|

and

the

initial

distribution

is

o(o)

=(0.7,0.2,2,0.1).

Find

[0.,

0.4

fi)

(t)

P

(x2

=

3)

(i1) P

(&

= 2,

x3

= 2,

x

2 = 3,

xt

= 3,

x

o =

2).

(oR)

b.

Three boys

A, B

and C are

throwing

a

bail to

each other.

A

always

throws

the

ball to

B

and

B

always

throws

the

bail to

C,

but

C

is

just

as

likely

to

throws

the

ball

to

B

as to

A.

Show

that the process is

Markovian. Find

the

transition matrix

and classify the status.

*****

Sample 1 18 13

t2

15

t2

l4

T6

l4

15

Sample 2

t6

t9

13 16 18

t3

15
(32)
(33)
(34)

ii.

Derive the moment generating fimction of

uriform

dishibution and hence firrd mean and variance

ofthe

distribution.

(oR)

b.i.

If

X

has

a

geometdc distribution,

then

for

any

two

positive

integers

m

and

q

prove

that

PIX>m+n/X>m]=PlX>nl.

ii.

Let

X

be

is

a

normal

variate

with

mean

30

and standard deviation

is 5. Find

ttre

following

Pl26

<

X

<

401,

plx

>

4sl

and

P1:lx

4oll>

5.

30.

a-

Two random samples gave the data.

SamDle Size Mean Variance

I

2

8

li

9.616.5

t)

2.t

Cao we conclude that the two samples have been drawn from the same normal population?

(oR)

b.i.

Test the significance

ofthe

difference between the means

ofthe

samples, dm\,,,rr from two normal populations with the same standarddeviation

SamDle Size Mean SD

I

2 100 200 61 63 4 6

ii.

The following data give the number

ofair-craft

accidents that occured during the various days

of

a

week-Days

Mon

Tues Wed

Tturs Fri

Sal

No. ofaccidents

l5

19

l3

t2

t6

15

Test whether the accidents are

unifomly

distribuGd.

31.

a.

A

deparlmental store has a single cashier. During the rush hours, customers araive at the rate of 20 customers per hour. The average number

of

customers that can be processed by the cashier is 24 per hour.

(i)

what

is

the probability that the cashier is idle?

(ii)

what is the average

number

of

customer in the queuing system?

(iii)

what is the aveEge time a customer spends in the system?

(iv)

whal is the

avemge number

of

customers

in

the

queue?

(v)

what

is the

average

time

a customer spends in ihe queue, waiting for service?

(oR)

b.

The local one-person barber shop can accommodate a maximum

of

5 people at a time (4 waiting and

I

getting hair cut). Customers ardve according to a Poisson distribution with mean 5 per hour.

The

baxber cuts

hair at an

avenlge

mte

of

4

per hour

(exponential service

time).

(i)

what

percentage of time is the barber idle? (ii) what ftaction

ofthe

potential customers are tumed away?

(iii)

what is the expected nunber

of

customers waiting

for

a hair

cufl

(iv)

how much time can a customer expected to spend in tl-re barber shop?

32.

a"

The transition probability matrix

of

a Markov chain

{\}:

n

= t,

2,

3...

having 3 states 1, 2, and 3

fo.r

o.s

0.4\

tt

is

-

P=l

tt

0.6

0.2

0.21.

Initial

distribution

is

P(0)

=

(0.7 0.2 0.1).

Find

(D

P{Xr=3}

(0.3 0.4

0.3,

(ii) P{x3=2,

&=3, xr=3,

&:2}

(oR)

b.

Three boys

A,

B

and

C

are throwing a ball

to

each other.

A

always thiows the

ball

to

B

and

B

always tl.[ows the

ball to

C, but C is

just

as

likety

to

tlrow

the

ball

to

B

as to

A.

Show

tlEt

the

process is Markovian. Find the tmosition*matrix and classi8, the states.

Reg. No.

B.Tech. DEGREE

EXAMINATION,

NOVEI\{BER

2015 Fourth Semester

.

MAIO14-

PROBABILITY AND

QI]ET]ING

TMORY

(For the candidates adnixed

d

ring ,he academic ))ear 2013

-

2014)

(Statistical tunb is to be provided)

Part - A should be a.nswered in OMR sheet within first 45 minutes ard OMR sheet should be ha.nded over to Note:

(,

hall iflvigilator ar the end of456 minute.

(i,

Part - B ard Part - C should be answered ifl arswer bookleL

Time: Three Hours

1.

Max. Marks: 100

PART

-A

(20

x 1=

20

Marks)

Answer

ALL

Questions

The probabiliry density

imction

ofa

contiruous random variable

is

f(x\:ce-'l;-a<x<a

then the va1u9

ofc

is

(A)

v2

(c)

3/4 dala.

z

altt

lf

"

/(r)=i

'

l0

(A)

0.33

:-

"-'-,

then

E(x)

is

(B)

1/4

(D)

1/6

(B)

(D)

(c)

2.33

3.

Yd44X+8) is

(A)

t2va{x)

(c)

t6ra4x)

1.33

(B)

4,za{x)+8

@)

161/a(x)+8

(B)

P

factorial moment

(D)

(r-l)d

cenaal moment

@)

Poisson

(D)

Normal

(A)

rd centlal moment

(C)

rd mw moment

1tt

(121=le,'1,,

-*.

r,.*

I

2Ja

5.

Which distribution has equal mean and variance?

(A)

Binomial

(C)

E).?onential

(A)

e-t

l;

x>o

(c) l-e-b;

x>o

4

IfX

is a random variable and r is an integer, then

E(-t

-t)'

represents

6.

Ifthe

probability that a target is destroyed on any one shot is 0.5, then the probability that it

would

be destroyed on 6d attempt is

(B)

0.0165

(D)

0.0156

(A)

0.106s

(c)

0.r056

7.

The cumulative distdbution fimction

off(x)

ofan

exponential distribution is

@)

"t'

-t;

x>o

@)

1-etu;

.r > o

8.

The prcbability density function of a standard normal distribution is

1

2l^

(ts)

0e\=-=e

'

t'.0<z<*

\l

zlt

1

)/^

(D)

0lzl=-=e' t'i-a

< z

<a

\lZJr

rct

drzt

=Le'zl,g.

r.*

"l2z

(35)

10.

The test used to test the equaliry ofvariance

oftle

populations &om two small samples is

(B)

F-test

(D)

Independent of altributes

20.

State

i

is said to be aperiodic with period

di,

if

(A)

di>

1

(B)

d'<1

(c)

4=1

(D) di=o

(A)

t-test

(C)

t'

.test

s/ ',J n

X-"

ll:l

-

7=

' sf

.ln

-r

In large samples, to test the sigrfficance

ofthe

difference between sample mean mean p, the test statistic (sample standard deviation s is known) is

al

,=P,

!

At

,=-!J:

slJn-l

.{

and population

21NA4nA10t,l

&=gardan

0<x<l

elsewhere

ulnown

probability

the

steady state

PART-B(5x4=20Marks)

Answer

ANY FIVE

Questio.s

A

random variable

X

has

a

mean

p

=

12 and variance distribulion. Find P(6<X<l 8).

22.

23.

25.

26. 21.

Given the random variable

X

with density fimction Find the

pobability

densiry

ofY

= 8X3.

A

favel

company has

two

cars

for hiring.

The demand

for

a cztl on each day

is

distributed as

Poisson variate with mean 1.5. Calculate the

propotion

ofdays on which some demand is refused'

24,

The

time (in hous) requied to

repair

a

maohine

is

exponentially distributed

with

parameter

L

=

U2. What

References

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