Reg.
No.
B.Tech.
DEGREE
EXAMINATION,
NOVEMBER
2016Third
Semester15MA207
-
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 withinfrst
45 minutes and OMR sheet should be handed over to hall invigilator at the end of 45m minute.(iD
Part - B andPart
- C should be answered in answer booklet.Time:
Three HoursMax.
Marks:
100PART
-A(20x1=20
Marks)
AnswerALL
Questions(
-x,/
l.
A RV X
has theprobability
densityfirnction
-f(*)=lk' ",
*
>0.
Thevalue of k
islo,
x<o
4'
Giu"n
E(X)
-1,
E(X\
=
2
fora
discreteRV X,
thenVar(3X+l)
is(A)
2(c)
I
Let
X
be a continuousRV
then(A)
F(-oo):
0(C)
F(m;
-
-1
Let
X
be a
discrete
RV
with
possible respectively. ThenE(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)
-
2values
1,2,3
associatedwith
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
isuniformly
distributed in
(-3,
3) then itsprobability
densityfunction
f(x)
isgiven by
6.
The normalprobability
curve
issymmetrical
about(B)
1
4
(D)
1
9(A)
1
6
(c)
1
3
Z.
If
theRV X follows
a Poissondistribution
with
mean|
,f,"r.rP(X:0)
isgiven by
2(A)
e-r
-1
(B)
e2
,
@)"2
^
(C)
ez8.
tf th" MGF
of adistribution
i"
"3(e'-1)
then the variance is(A) 1
(B)
1(D)
3(c)
e9.
The Chi-square test is used totest
(A)
The
difference between population
(B)
The
difference between
population
means
variances(C)
The
goodnessof
fit
(D)
The difference betweenproportions
1
0.
If
a researcher rejects anull
hypothesiswhich
is true thenit
is a(A)
Type
1error
(B)
Type 2error
(C)
TypesA
error
(D)
Standarderror
11.
The
't'
distribution
tends to_
distribution
for sufficiently
largevalue
ofy
(A)
Uniform
(B)
ExPonential(C)
Standardnormal
(D)
Poisson12.
_is
usedto
test thedifference
between the population variances(A)
't'-test
(B)
Chi-Square test(C)
Z-test
(D)
F-test13.
Thesymbol
c inthe
queueingmodel
(alblc): (dle) standsfor
(A)
Systemcapacity
(B)
Number of
servers(C)
Queuesize
(D)
Queuediscipline
14.
In
a queueing system thearrival
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 behaviorof the
queueing system doesnot
depend ontime,
then the system is saidto
bein
(A)
Transientstate
(B)
Idle
state(C)
Steadystate
(D)
Busy
state16.
If
)"-8
/ hr
andE(N)
-
4
customersfor
(MlMll):(mlFIFO)
queue systemthen
p is givenby
(A)
e
(B)
10(c)
1r
(D)
1217. A Markov
chainis
said to bea-periodic
if
(A)
dr=l
(B)
di=Z
(c)
di:o
(D)
3Page 2
of5
2INF315MA2079
18.
Irthe
tpmora
Markov
"n"t"
t,
[fi
irf""or.,
=(;,+)
thenptrri,
1
19.
The reactionto
statei
is uncertainif
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 thanI
(A)
Greaterthan
1(c)
120. A
Markov chain
is saidto
be absorbingif
it
has one absorbing rate@)
Atleast
(D)
Nearly
PART-B(5x4=20Marks)
Answer
AI{Y FIVE
Questions21.
Theprobability
distribution
ofX
isFind
the mean and variance ofX.
Buses
arrive at
aspecified
stopat
15min
intervals
startirrgatl
A.M,
that
is theyarrive
at 7,7;15, 7:30,
7:45
andso on.
If
a
passengerarrives at the stop
at a
random
time
that
isunifonnly
distributed
betu,een7
and7:30
A.M. Find
theprobability
that
hewaits atleast
12min
for
a bus.A
sampleof
size
13 gave an estimatedpopulation
varianceof
3.0,while
another sampleof
size
15gave an
estimate
of
2.5-
Could both
samplesbe
fiom
populations
with
the
same variance?A
student'sstudy
habit's
are asfollows:
If
he studies onenighthe
is 70%
surenot
to
studythe
nexlnight.
On the other hand,if
he doesnot
study onenight,
heis 60%
surenot
to
study the next aswell.
Write
the tpm of theMarkov
Chain.Explain
thesymbolic
representationof
a Queuing model.Given
theRV X
with
densitytunction
f
(x)
={?.'
l '] '
I
n,ra theof
Y:
8X3. 10, elsewhereState and
prove
memoryless properfyof
exponentialdistribution.
22.23.
24.
25.
26.
27.
x:
0 .,, 4 6P(x):
I
6
I
;
J 8I
:
3E
PART-C(5x12=60Marks)
Answcr
ALL
Questions28.
a.i. A
continuousRV
X
has(x):k
(l-x)
for
0 <
x <
l.
Find
the rth moment aboutthe
origin.
Hencefind
mean, variance.ii.
If
X
denote thenumber in
athrow
of
afair die furd
E(X),
E(9x+2), Var
(X).
(oR)
b. A
fair
die
is tossed600 times. Use TchebychefPs inequality
to
find
a
lower
bound
for
the
probability ofgetting
80to
120 sixes.29.
a.
Find
theMGF
ofBinomial
distribution
and hence frnd the mean and variance.b.
In
anormal
distribution,
\Yoof ther*::?"
under
35
and \9o/oareunder 63. What
arethe
mean and standarddeviation of
thedistribution?
30.
a. A
group
of
5 patients treatedwith
medicineA
weigh
42,39,48,60 and 41kg,
a secondgroup
of
7 patients
from the
samehospital
treatedwith
medicine
B
weigh
38,42.56,64,68,69 and62.Do
you
agreewith
theclaim
that medicineB
increases theweight significantly?
(oR)
b.
In
a
locality
100
persons
were
randomly selected and asked about
their
educational achievements. The results aregiven
asEducation
Middle
High
school College
Total
Sex
Male l0
15
25
50Female
25
10
15
50Total 35
25
40
100Can
you
say that education depends on sex?31.
a.
Arrivals
at
a telephonebooth
are consideredto
be
Poissonwith
an
averagetime
of
10min
betw,een onearrival
and the
next.
The length
of
a phone
call is
assumedto be
distributed
exponentially with
mean 3min
(i)
Find the average numberof
personswaiting
in the system(ii)
What isthe
probabiliff
that
a personarriving
atthe booth
will
haveto wait in
the queue?(iii)
What isthe
probability
that
it will
takehim
morethan
10min
altogether towait for
phone and complete
his
call?(iv)
The
telephone department
will
install a
secondbooth when convilced that
anarrival
hasto wait
on
the
averagefor
atleast3
min for
phone.By
how
much
theflow
of arrivals
should increasein
order tojustify
a second booth?(oR)
b.
In
a
single
serverqueuing
systemwith
Poisson
input
and exponential service times.
If
the meanarrival
rate is3
calling units
per hr, the mean service rate is4
perhr
and themaximum
numberof
callingunits
in the system is 2,find
(i)
P,(n>O)
(iD
Averagenumber of calling
units in the system(iii)
Averagewaiting
time in
the syslem(iv)
Averagewaiting
time in
the queue.lo.z
0.332.u
TheQmof
aMarkov
chatn{X^,n>0}havingthree
states0,
I
and,
t,
"=
|0.,
0.6Lo.o
0.3 Theinitial
distribution
is givenby 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'sterritory
consistsof 3 cities
A, B
and
C.
He never sells
in the
samecity
on successive days.If
he sells
in
crty
A,
then the
next
day
hesells
in
B.
However,
if
he
sellseither in
B
of
C,
thenthe next
day heis
twice
aslikely to
sell
in city
A
asin
the
othercity.
How often
does he sellin
eachofthe
cities
in the steady state?0.sl
031,
031
ii.
ln
a normal distoibution, 7Yoof
the items are under 35 and 89% are under 63. What are the mean and standard deviation of thedistribution?
30.
a.i.
kr
a largecity
A,
20%of
a random sampleof
900 school boys had a slight physical defect.In
anotherlarge
city
B,
18.5%of
a
random
sampleof
1600school boys had
the
same defect. Is the difference between the proportions significant?ii'
The mean height and the SDheight
of
8raadomly
chosen soldiers are 166.9 and 8.29 cms respectively. The corresponding valuesof6
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
apolitical
proposal.
Of
them, 1872 were men and
tle
rest women.
A
total
of
2ZS7individuals were'in favour of
the proposal and 917 were opposedto
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.
Customersarrive at
a
one-man barber shop accordingto
a Poisson processwith
a mean interarrival
timeof
12min.
Customers spend an averageof l0 min
in the barber's chair.(1)
What is the expected number of customersin
the barber shop and in the queue?(2)
Calculate the percentageof time
anarrival
canwalk
straight into the barber's chairwithout
having towait.
(3)
How
muchtime
can a customer expectto
spend in the barber's shop?(4)
What is the averagetime
customers spendin
the queue?(5)
What is theprobability
that thewaiting time in
the system is greater than 30 min?(6)
Calculatethe
percentageof
customerswho
haveto wait prior to
getting
into
the barber's chair,(oR)
b.
The
local
one-person barber shop can accommodate andmaximum
of
5
peopleat
a time(4 waiting and
1getting
hair-cut).
Customersarrive
accordingto
a
Poissondistribution
with
mean 15 perhour.
The
barber cutshair
at an average rateof4
perhour
(exponential servicetime)
(l)
What percentage of time is the barber idle?(2)
What fractionof
the potential customers are tumed away?(3)
What is the expected numberof
customerswaiting for
a hair-cut?(4)
How
muchtime
can a customer expectto
spend in the barber shop?32.
a.
A
man either drives a car or catches atrain to
goto
oflice
each day. He never goes2
daysin
arow by
train
but
if
he drives
one day, thenthe next
dayhe
isjust
aslikely to
drive again as heis to
travel by
train.
Now
supposethat on
thefirst
day of
the week, the man tossed afair
dice and drove towork
if
andonly
if
a 6 appeared.Find
(i)
theprobability
that he takes a train on thethird
day(ii)
theprobability
that he drives towork
in the long run.(oR)
b.
Three boysA, B
andc
aretfuowing
aball
to
eachother.
A
alwaysthrows
theball to
B
andB
alwaystfuows the ball
to
c,
but
c
isjust
aslikely
to
throw
theball
to
B
asto
A.
Show that the process isMmkovian.
Find the transitionmatrix
and classifu the states.Reg.
No.
B.Tech.
DEGREE
EXAMINATION,
MAy
2012Third /
Fourth SemesterI5MA2O7
_
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 handedover 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 variableX
has theThe value
of
K
is.(A)
2/24(c)
7/122.
var[+x
+g]
is(A)
rzvar[xj
(c)
t6var[x]
PART-A(20x1=20Marks)
Answer
ALL
QuestionsMax.
Marks:
1003.
When a die ist}rown, X
denotes the number that tums up,find
the mean@)
2U24
@)
1/24(B)
avar[x]+s
(D)
16var[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 PDFof
a random variableX
is/(x)=Zx; 0<x<l,find
p(,Y>0.5)
(B)
2t3(D)
4/s
(A)
1/2(c)
3/45.
If
on an averagg 9 ships outof
10arrive
safelyto
apor!
then mean andsD
of the numberof
shipsreturning
safely outof
150 ships are(A)
13s,2.674
(c)
13s,3.674
Mean of the Poisson
distribution
is(A)
l.
(c)
1/tu7.
TheMGF of
Geometricdistribution
is(A) 1
(B)
l-q"'
(B)
?,+1(D)
T,
(D)
J!-l-q"'
function
0
I
2 3 4P(x): k 2k 5k 7k 9k
PaEe 4 of 4
22MF3l4l5Il,A207
pageI of4
(c)
L-
pe8.
The variance ofUniform
distribution
U(a,b)
is20. A
state'i'
is said to be aperiodicwith
period'di'
if
(A)
I
(,r-
r)'
12'
'
(c)
1(a*o)
2'
9.
The value setfor
cr isknown
as(A)
Rejection level(C)
Signihcance level(B)
(D)
Lr@-")'
)o-,)
(B)
Acceptancelevel
(D)
Hypotheti callevel
(B)
Standardenor
@)
Sampleenor
@)
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:
oPART-B(5x4=20Marks)
Answer
ANY FfVE
Questions10.
The standard deviationofa
samplingdistribution
is called asA
continuous random variablex
has a/(x)
=laze-';
x>
0 ' Find the valueof
k'
Given the random variable xwith
densityfunction
l2x
o<x<1
/(x)
=.{-"
. Find the pdf ofY
:
8X3.r
\ '' L0
elsewhereIf
theprobability
that an applicantfor
adriver's
licensewill
pass the road test on arry.giventrial
iJ 0.8, what is theprobability
that hewill
finally
pass thetest (i)
on thefourth
hial
(ii)
in
fewer than 4trials.
The
mileagewhich
canowner
getwith
acertain
kind
of
radial
tire is
a random variable having an exponentialdistribution with
mean 40, 000 km. Find the probabilities that oneof
these tireswill
last(i)
atleast 20,000kms(ii)
atrnost 30, 000kms.Define
(i)
null
hypothesis(ii)
altemate hypothesis(iii)
Bpe
I
error(iv)
typeII
error.what
do
the letters
in the
symbolic
representalion (a/b
Ic):(d
Ie)
of
a
queuing model represent?27.
Ifthepmof
aMarkovcirainis
[,:"
.
]".l
Findthesteadystatedistributionof
thechain.
11/2
1t2)
2t.
(A)
Sampling error(C)
Simple error11.
A
is a subsetof
a23.
24.
12.
The hypothesis that anaralyst
istrying 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)
Ergodic13.
The symbolic notationof
queuingmodel
is representedby
(A)
Kendall
(B)
Euler
.
(C)
Fisher
(D)
Neuman14. The interval
betweentwo
consecutive arrivals
of
a
Poissonprocess 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.
Theprobability
of
'n'
customers in the system Po:
,o,
[4)r.
\p)
'"'
(#)'",
16.
Which
term refers to"a
customerwho
leaves the queue because the queue istoo long"
l)
,0
Po
(
s"\
ttlp)
(+)
PART-C(5x12:60Marks)
Answer
ALL
QuestionsA
random variableX
has thefollowing probabilit
x:
|\-1
0I
2 3P(x): 0.1
k
0.22k
0.33k
distribution
(i)
Find
k (ii)
evaluateP(X
< 2)
(ii|
P(-2
<
X
<
2)
(iv)
CDF of
X
(vi)
variance ofX.
(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 ), usingTchebycheff
s inequality.If
X
represents the outcome,when
afair
dieis
tossed,furd
theMGF
of
X
and hencefind
E(X)
and var(X).
out of
800 familieswith
4 children each, how many familieswouid
be expected to have(i)
2
boys
and2 girls
(ii)
atleastI
boy (iii)
atmost2 girls
(iv)
children
of both
sexes. Assume equalprobabilities for
boys and girls.(oR)
Buses arrive at a specified stop
at
15 minsinterval
starting at 6 am(ie)
arrive at 6am' 6.15am, 6.30
am and
so
on.
If
a
passengerarrives at
the
stop
at a time that is uniformly
distributed
between6
and 6.30 am,find
theprobability
that hewaits
(l)
less than5
minsfor
a bus (2) morethan
10 minsfor
abut.
,2MF3/4r5r\ra207
17.
The sum ofall
the elementsof
anyrow
of
the hansitionprobability matrix
is(A)
0(c)
0.7s(B)
0.s(D)
I
The
steady stateprobability
vectorr
of
a discretemarkov
chainwith
transition probability
matrix
satisfies thematrix
equation.18.
19. A
non-null persistent and aperiodic state is calledb.i.
on
Ior
lneIollo
0
.r 1 2 3 4 5 6 Total
f
5 18 28t2
-, 6 4 80ii.
A
continuous random variableX
has af
(x)
=b2
"-*
,
frndh
mean and variance.29. a.
Fit
abinomial distribution
for
thefoll
data:30.a.
(oR)
In
an engineeringexaminatioq
a studentis
consideredto
havefailed,
secured second class,fust
classand
distinction
according
ashe
scores lessthan
45olo,between
45Yo artd 600/o,between 60% and 7 5%o and above 7 5o/o respectively.
In
aparticular year
l0o/o of the studentsfailed
in
the
examination and
5o/o of.the
studentsgot
distinction.
Find the
percentagesof
students
who
have gotfrst
class and second class.15.5 percent
of
a random sampleof
1600 undergraduateswere
smokers, whereas 20Voof
arandom
sampleof
900 post-graduateswere
smokersin
a
state. Canwe
concludethat
lessnumber
of
undergraduates are smokers than the postgraduates?(oR)
Theory predicts that the proportion of beans in four groups
A, B,
C andD
should be 9:3:3:1. In anexperiment among 1600 beans the numbers
in
the four groups were 882, 313, 287 arrd 118. Doesthe experiment support the theory?
Arrivals at a
telephonebooth
are
consideredto
be
Poissonwith
an
averagetime
of
12minutes between one
arrival
and
the next. The length
of
a
phone
call is
assumedto
be distributed exponentiallywith
mean 4 minutes.(i)
Find the average number of personswaiting
in the system.(ii)
What
is theprobability
that a personarriving
atthe
boothwill
haveto
wait
in
the queue?(iii)
Estimate thefraction
of the day when the phonewill
be in use'(iv)
What
is
the
probability
that
it
will
take
hirn
more
than
10minutes
altogether towait for
the phone and complete his call?31. a.
Reg. No.
B.Tech.
DEGREE
EXAMINATION, MAY
2018 First to Sixth Semestert5MA207
-
PROBABILITY AND
QUEUING TFIEORY(For the candidates adnitted during the academic year 2015
-
2016 omtords) Note:(i)
Part -
A
should be answeredin
OMR sheetwithin first
45 minutes and OMR sheet should behanded over to hall invigilator at the end of 45d minute.
(iD
Part - B andPart
-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 thenE(c)
is(A)
0(c)
cf(9)2.
Var(4X
+8)
is(A)
l2Var(X)
(C)
16Var(X)
3. A
randomvariable
X
hasmean
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 Poissondistribution
is(A)
r"(C)
I'
7.
TheMGF of
geometricaldistribution
is(A)
1l-
q"'
(c)
ql-
pr'
Page 1 of 4
(B)
4Var(X)+8
(D)
l6Var(X)+8
12
and varianced:9
and(B)
p
=npq,oz
=np
(D)
p
-np,
o2
=pe
(B) l.+
t
(D)
x-l
(B)
1l-
pu'
@)
pet7-
qu'an unknown probability
(B)
I
(D)
c b.(B)
1
4@)1
8
(oR)
b.
The local
one-personbarber
shop can
accommodatea maximum
of 5
people
at
a
time(4
waiting
and
lgetting
hair-cut).
Customersarrive
accordingto
a Poissondistribution
with
mean 5 per hour. The barber cuts hair at an average rate
of4
per hour.(i)
What percentageof time
is the barber idle?(ii)
Whatfraction
of the potential customers are tumed away?(iir)
What is the expected numberof
customerswaiting
for
a hair-cut?(iv)
How
muchtime
can a customer expectto
spend in the barber shop?32.
a. A
gambler hast2
he bets{1
at atime
andwins
{1
with
probabiliLy
112.He
stopsplaying
if
he loses T2 or wins{4.
(i)
What is thetpm of
the relatedMarkov
chain?(ii)
What is theprobability
that he has lost his money at the endof
5 plays?(iiD
What is theprobabiliff
that the game lasts more than 7 plays?(oR)
b.
Three boysA,
B
and C arethrowing
a ballto
each other.A
always throws the ball toB
andB
always throws the ballto
C, but C isjust
aslikety
tothrow
the ballto
B
as toA.
Show that the process inMarkovian.
Find the transitionmatrix
andclassiff
the states.:F****
4'
rcn(xz)
-8
and
E(X)
-2,then
Var(X)
is5.
The mean and varianceof
abinomial distribution
is(B)
2(D)
4(A)
3(c)
119.
If
one step transitionprobabiliry
does not depend on the step, then theMarkov
chain is called8.
If X
isuniformly disfibuted
in
(0,
10), thenP(X >
8) is(A)
Reducible(C)
Homogeneous20.
The
steady stateprobability
vector
matrix
equation(A)
nP:0
(C) nP:x
(B)
Regular(D)
Non homogeneousz
of
A
discrete
Markov
chain
with
TPM
satisfies the(B) zP:p
(D)
n(l+P)
-
02,3and4such
probability
distribution
(A)
r/s
(c)
3/59.
Theform
of the altemative hypothesis can be(A)
One-tailed(C)
Neither
one nortwo
tailed(A)
Samplingenor
(C)
Standard error10.
What is the standard deviationof
a samplingdistribution
called?(B)
r/ro
(D)
1/3@)
Two-tailed
@)
One ortwo tailed
(B)
Sample error(D)
Simple error12.
Student'st-distribution
has(n-1)
sample are(A)
Dependent(C)
Maximum
13.
What standfor
'd'
in the queuemodel
(a I b Ic:d
/ e)(A)
Queue discipline(C)
Servicetime
degrees
of
freedom. When
all
the
n observationsin
the(B)
Independent(D)
Minimum
ffi,-.3,$;fl;1ffim
If X
is uniformly distributedt f+,+)
find the pdfof
.l'=tanX
\2'2
)
If
theprobability
that an applicantfor
adriver's
licensewill
pass the road test on anygiven
trial is
0.8, what is theprobability
that hewill
finally
pass the test(i)
on thefourth
trial
and(ii)
in
fewer than 4trials?
The
fataliq,
rateof typhoid
patientsis believed
to be
l7.26percenl In
a
certain yeat
640patients suffering
from
typhoid were
treatedin
metropolitan hospital and
only
63
patients died. Can you consider the hospital efficient?In
the usual notationofa(M
lM
/l):(a/
FIFO)
queue systemifX:
12perhour
andp:24
per hour,
find
the average numberof
customers in the system and in the queue.The
transition
probability
matrix
of
aMarkov
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 truenull
hypothesis, a error occurs.(A)
Type
I
(C)
TypeB
@)
TypeA
(D)
Typetr
z3-24.
25.
t4.
p-1"
(c)
x
Lr+I
(A)
Balking
(C)
Jockeying(A)
0.18(c)
0.38(A)
Present(C)
Future18.
Ergodic means(A)
Ineducible
andperiodic
(C)
Not
ineducible
(B)
System capacity(D)
Numberof
serversl"-p
(D)
p
r+p
(B)
Reneging(D)
Leaving
(B)
Past(D)
Outcome(B)
kreducible
and aperiodic(D)
RegularThe average number
of
customersin
the systemn
(M
IM
/l:
a
IFIFO)
model is(A) l,
(B)
p
15.
Whichterm
refersto
oa customer.who leaves the queue because the queue istoo long'?
If X
is
a randomvariable
whose densityfunction
i,
"f(x)
= €--x ,-0 <r
< co.Find
rth
moment-
0 otherwise aboutorigin
andhencefind
p1and
12.
State and prove memoryless property of the exponential
distribution.
16. In (M
IM
/l:K
/FIFO)
model,if
)":3
perhour,
pt:4
perhour
andeffective
meanarrival
rate
of
a customer is 2.88 per hour then what Pn?26.
a1
(B)
0.28(D)
0.4817.
Markov
proves is onein
which the future value is independentof
values.PART-C(5x12=60Marks)
Answer
ALL
Questions28.a.
If
the
random
variable
X
takes
the
values
1,2P(X
-l)=3P(X
-2)=
P(X
=3)
-sP(X
-4),
find
the cumulativedistribution function of
X.
that
and(oR)
b.i
A
fair
die is
tossed720
times.Use
TchebychefPsinequality
to find
a lower
boundfor
theprobability
of getting 100to
140 sixes.ii.
Derive the moment generating fimction ofuriform
dishibution and hence firrd mean and varianceofthe
distribution.(oR)
b.i.
If
X
hasa
geometdc distribution,then
for
any
two
positive
integersm
and
q
prove
thatPIX>m+n/X>m]=PlX>nl.
ii.
Let
X
beis
a
normal
variatewith
mean30
and standard deviationis 5. Find
ttrefollowing
Pl26
<X
<
401,plx
>
4sland
P1:lx4oll>
5.
30.
a-
Two random samples gave the data.SamDle Size Mean Variance
I
2
8
li
9.616.5t)
2.t
Cao we conclude that the two samples have been drawn from the same normal population?
(oR)
b.i.
Test the significanceofthe
difference between the meansofthe
samples, dm\,,,rr from two normal populations with the same standarddeviationSamDle Size Mean SD
I
2 100 200 61 63 4 6ii.
The following data give the numberofair-craft
accidents that occured during the various daysof
a
week-Days
Mon
Tues WedTturs Fri
SalNo. ofaccidents
l5
19l3
t2
t6
15Test 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 numberof
customers that can be processed by the cashier is 24 per hour.(i)
whatis
the probability that the cashier is idle?(ii)
what is the averagenumber
of
customer in the queuing system?
(iii)
what is the aveEge time a customer spends in the system?(iv)
whal is the
avemge numberof
customersin
the
queue?(v)
whatis the
averagetime
a customer spends in ihe queue, waiting for service?(oR)
b.
The local one-person barber shop can accommodate a maximumof
5 people at a time (4 waiting andI
getting hair cut). Customers ardve according to a Poisson distribution with mean 5 per hour.The
baxber cutshair at an
avenlgemte
of
4
per hour
(exponential servicetime).
(i)
whatpercentage of time is the barber idle? (ii) what ftaction
ofthe
potential customers are tumed away?(iii)
what is the expected nunberof
customers waitingfor
a haircufl
(iv)
how much time can a customer expected to spend in tl-re barber shop?32.
a"
The transition probability matrixof
a Markov chain{\}:
n= t,
2,3...
having 3 states 1, 2, and 3fo.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 boysA,
B
andC
are throwing a ballto
each other.A
always thiows theball
toB
andB
always tl.[ows theball to
C, but C isjust
aslikety
totlrow
theball
toB
as toA.
ShowtlEt
theprocess 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]INGTMORY
(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 bookleLTime: Three Hours
1.
Max. Marks: 100
PART
-A
(20x 1=
20Marks)
AnswerALL
QuestionsThe probabiliry density
imction
ofa
contiruous random variableis
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:-
"-'-,
thenE(x)
is(B)
1/4(D)
1/6(B)
(D)
(c)
2.333.
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 moment1tt
(121=le,'1,,
-*.
r,.*
I
2Ja5.
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, thenE(-t
-t)'
represents6.
Ifthe
probability that a target is destroyed on any one shot is 0.5, then the probability that itwould
be destroyed on 6d attempt is(B)
0.0165(D)
0.0156(A)
0.106s(c)
0.r0567.
The cumulative distdbution fimctionoff(x)
ofan
exponential distribution is@)
"t'
-t;
x>o
@)
1-etu;
.r > o8.
The prcbability density function of a standard normal distribution is1
2l^(ts)
0e\=-=e
'
t'.0<z<*
\l
zlt
1
)/^
(D)
0lzl=-=e' t'i-a
< z<a
\lZJrrct
drzt
=Le'zl,g.
r.*
"l2z
10.
The test used to test the equaliry ofvariance
oftle
populations &om two small samples is(B)
F-test(D)
Independent of altributes20.
Statei
is said to be aperiodic with perioddi,
if
(A)
di>
1
(B)
d'<1
(c)
4=1
(D) di=o
(A)
t-test(C)
t'
.tests/ ',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) isal
,=P,
!
At
,=-!J:
slJn-l
.{
and population21NA4nA10t,l
&=gardan
0<x<l
elsewhere
ulnown
probabilitythe
steady statePART-B(5x4=20Marks)
Answer
ANY FIVE
Questio.sA
random variableX
hasa
meanp
=
12 and variance distribulion. Find P(6<X<l 8).22.
23.
25.
26. 21.
Given the random variable
X
with density fimction Find thepobability
densiryofY
= 8X3.A
favel
company hastwo
carsfor hiring.
The demandfor
a cztl on each dayis
distributed asPoisson variate with mean 1.5. Calculate the
propotion
ofdays on which some demand is refused'24,
Thetime (in hous) requied to
repair
a
maohineis
exponentially distributedwith
parameterL
=
U2. What is
the conditionalprobability
thata
repair takes a.tleast 10hous
giventhat
its duration exceedst
hou$?Experience has
shoun thal
20'/oof a
manufactured productis
of
top
quality.
In
one
day's productionof
400 articlesonly
50 areof
top quality. Show that either the productionof
the daychosen was not a reprcsentative sample or the hypothesis of 20% was wrong.
In
a railway marshalling yaxd goods trains aEive at rateof
30 trains per day. Assuming that the inter-arrivai timefollowi
an exponential distribution and the service time is also exponentialwith
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 natureofthe
(A)
Null
hypothesis(B)
Altemative hypothesis(C)
Levelofsignificanoe
(LOS)
@)
Degreesoffieedom
13. In
queuing system, the mrmbetof
customers sewiced perunit
has a Poisson distributionwith
mean(A)
p(c)
2tv
In the symbolic representation
ofa
queue model (a I b /c):(d
/ e), wlrere c denotes.(A)
The typeofdishibution
ofthe noof
(B)
The typeofdishibution oflhe
no ofservicest
).
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/pper rmit time
(D)
The capacityofthe
system(B)
Two-step transitionprobability(D)
n-step transition probability@)
ri1<0
&l
Pii
=l
@)
eii
<0
&f
P,t=1
(B)
Ineducible@)
AperiodicArrivals
at a telephone booth are considered to the Poissonwith
an averagetime
of
12 minutes betweenone arrival
andthe
next.The length
of
a
phonecall is
assumedto 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)
2.016.
Probabititytlat
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
eo17.
The
conditionalprobability
that the4
at step 0 is called(A)
One-steptrarsitionprcbability(C)
(n-
1) step tra.sition probabilityprocess
is
in
stateaj at
stepq
giventhat
it
wasin
state18.
The P',r of a Markov chain is stochastic matrix, becausePART-C(5x12=60Marks)
Answer
ALL
Questions 28.a.i.
A mndom vadable X has thefo
(A)
Find
K
(B)
EvaluateP(X<
2)
(C)
EvaluateP(-2<X4)
@)
Find the
cumulative distribution fimctionofx
(E) Evaluate the mean and varianceofx.
. Ihe
probability
tunctionb.r.
Ptx
=
i)
=+lj
=1,2,....@). 2rofthe
disaibution.I
-21.ii.
rf
f(
=|xe
"
/':
r>u
[0
I
x<0
(l)
Show thatf(r)
is a EobabilirydeNity
function of a continuous random variableX
(2)
Find its distribution F(.t).19.
tf
pl"!o
for
somen
andfor all
i
andj
then every state can be rcachedfiom
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 probabilityis
1 and find the mean and variance21NA4MAI014
Reg.
No.
B.Tech.
DEGREE
EXAMINATION, MAY
2018Fourth
SemesterMA1014
-
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 withinfrst
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 HoursMax.
Marks:
100PART-A(20x1=20Marks)
Answer
ALL
Questions1.
The
probability density'
function
of a
continuous
random variable
isf
(*)
-
C"-l*l,-
co <r
<€
then the valueof
C:
(A)
1
2
(c)
1
4
!=4x+3
is(A)
]r'-tl
(c)
)O_rl
(A)
22-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)
33-t
(D)
\z_t)-2
and
varian"".o2
=9
and an unknown
probability
1
4T
6
2.
If
the
continuous
random
variable
X
has
/(x)
={:*'o<x<l.
rhen
the
pdf of
' [0,
otherwise(B)
frfr_rl
(D)
)tr_rl
3.
A
randomvariable
X
has thegiven
by
f
(*\={'"-'*,'=',
thenMGF
isLo,
x<o
(B)
I
4(D)
1
8
5'
If
X
is
a Poisson variate suchthat
,(r')
=
o ,t.r-,
r(x)
is(B)
2(D)
06.
The mean of theexponential distribution
with
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)
18.
Normal distribution
is the
limiting form
of
conditions.(A)
Binomial
(C)
Geometricdistribution
under suitable statistical
9. The Chi-square goodness
of
fit
test can be used to testfor
(A)
Significanci of
samplestatistics
(B)
Difference
betweenpopulation
means(C)
Normality
(D)
ProbabilitY
Which
of
thefollowing
values isnot
typically
usedfor
o?
(B)
0.0s(D)
0.2s(B)
Poisson(D)
Uniform
(B) n-2
(D)
n-l
(B)
TypeA
(D)
TypeB
(B) 71
--r
u(D)
)"
,-+r
u10.
(A)
0.01(c)
1.101
1.
The degrees of freedomfor
t test based onn
observations is(A)
Zn-r
(c)
z(n-r)
12.
When the researcher rejects a truenull
hypothesis aerror
occurs?(A)
Type
I
(C)
Type
II
13.
Theprobabiliry
ofno
customersin the
systemi"
(M
IM
ll;a
IFIFO)
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)
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
statei
is saidto
be aperiodicwith
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
nonnull
persistent andaperiodic
state is called(A)
Eorpty
(C)
Ergodic
19.
The statei
is
said to be nonnull
persistentif
its mean recurrencetime
pii
is20.
The
steady stateprobability
vector
z
of
a
discreteMarkov
chain
with
transition probability
matrix p
satisfies thematrix
equation2t.
22.
PART-B(5x4=20Marks)
Answer
All-Y
FM Questions
?
If
a randomvariable
X
has moment generatingfunction
ltr*lt)
=*.
Obtain the
standarddeviation of
X.
Trains arrive
at a stationat
15 minutesintervals starting at
4am.If
a passengerarive
to
the statin at atime that
isuniformly
distributed
between 9.00 and 9.30find
theprobability
that
he has towait
for
thetrain for (i)
less than 6minutes
(ii)
morethan
10 minutes.Theory predicts that the
proportion of
beansin four
groupsA, B,
C,D
should be 9:3:3:1.In
an
experiment
among 1600 beans,the
numbersin
the four
groupswere 882,
313,287
and1i8.
Doesthe
experiment support the theory?Explain
Kendall's
notation for
queuing models.State and
prove
memorylessproperly of
geometricdistribution.
Find
MGF
of
the random variable whosemoments
arepr':
(r
+1)3'
andfind
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.
PART
-
C (5
x
12 -- 60Marks)
Answer
ALL
Questions 28.a.i
If
X
is randomvariable
having the densityfunction
^,
\
fxl6,x=1,2,3
,r(r)
={0,
otherwiseFind
E(f
\
+2x+Z)
/
anOvar(4x+5).
\
"
(8Marks)
ii.
If
the
continuousrandom variable
X
has
f
(x)=2/9(x+t),-t
<x<2
find
the
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
densityfunction,
meane varianceof
X.
(8 Marks)ii.
A
fair die
is
tossed 720times. Use Tchebycheff
inequality to
find
a
lower
bound
for
theprobability
ofgetting
100to
140sixes.
(4 Marks)29.
a.i
Fit
a Poissondistribution for
onlor
thefoll
datax
0 I 2 J 4 5Total
f
t42
156 69 27 5 1 400Find the
probability
massfunction
and thenfind
thetheoretical
frequencies.
(8 Marks) Thetime (in
hours)required
to
repair
a machineis exponential distributed
with
parameter )"=l
I2.
What
is the
conditional
probability
that a
repair
takesatleast
10hgiven that
its
duration exceeds
th?
(4 Marks)(oR)
b.i
In
an engineeringexamination,
a studentis
consideredto
havefailed,
secured second class,frst
class
anddistinction,
according
ashe
scoresless
that
45yo,between
45%
and 60%
between 60%
and 75Yo and above
75o/orespectively.
In
a particular
year
10%
of
thestudents
failed
in
the
examination
and
5%
of
the
students
got
distinction.
Find
the percentagesof
studentswho
have gotfirst
class and second class. (8 Marks)ii.
If
theprobability
that
an applicantfor
adriving
licensewill
passthe
road teston
anygiven
trial
is
0.8. What is theprobability
that hewill
finally
pass the test on thefourth trial?
(4 Marks)
30.
a.i
Random samplesof
400 men and 600women
were asked whetherthey
would like to
have a school neartheir
residence 200men
and 325 womenwere
in favour
of
theproposal.
Test the hypothesis thatthe proportions
of
men
andwomen
in
favour
of
theproposal
are same,at5%
level of
significance. (8 Marks)ii.
A
group
of five
patients treatedwith
the medicine
'A'
weights
42,
39,
48,60
and4l
kg;
u secondgroup
of
7 patients
from
the
samehospital treated
with the medicine
'B'
weights
38,42, 56,64,68,
69
and 62kg. Do you
agreewith
the claim
thatmedicine
B
increases theweight significantly?
(4 Marks)(oR)
b.i
The
mean
of
2large
samples 1000 and
2000
members
are 67.5
inches
and 68
inchesrespectively.
Canthe
samples be regarded asdrawn
from the
samepopulation
of
standarddeviation:2.5
inches? (8 Marks)ii.
Two
independent samplesof
sizes9
and7 from
a
normal
population had
the following
valuesof
the variablesDo
the estimates of thepopulation
variancediffer significantly
at 5Yo level? (4 Marks)31.
a.
Customersarrive
at a watchrepair
shop accordingto
a Poisson process at a rateof
one perevery
10 minutes and servicetime
is an exponential random variablewith
mean 8 minutes.(i)
Find
the average numberof
customersLs
in
the shop.(iD
Find the
ayeragetime
a customer spendin
the shop Ws.(iii)
Find
the averagetime
that a customers spendsin the
queue Wq.(iv)
Find the
average numberof
customersin
the queueLq.
(v)
What
is theprobability
that the serveris
idle?(oR)
b.
In
a car wash servicefacility,
carsa:rive
for
service accordingto
Poissondistribution
with
mean 5 perhour. The
time for
washing
andcleaning
each car hasexponential distribution
with
mean 10 minutesper
cax.The.facility
cannot handlemore
thanone
car at atime
and hasatotal of
5parking
spaces.(i)
Find
theeffective arrival
rate.(ii)
What is the probability
that an
arriving
car
will
get
service immediately
uponwrival?
(iii)
Find
the expected numberof parking
spaces occupied.32.
a.
Thetransition
probability
matrix
of
aMarkov chain
\Xr\,n--I,2,3...
having
3 states 1, 2(r.,
o.s
oo)
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 boysA, B
and C arethrowing
abail to
each other.A
alwaysthrows
theball to
B
andB
alwaysthrows
thebail to
C,but
C
isjust
aslikely
tothrows
theball
to
B
as toA.
Show
that the process isMarkovian. Find
thetransition matrix
and classify the status.*****
Sample 1 18 13
t2
15t2
l4
T6l4
15Sample 2
t6
t9
13 16 18t3
15ii.
Derive the moment generating fimction ofuriform
dishibution and hence firrd mean and varianceofthe
distribution.(oR)
b.i.
If
X
hasa
geometdc distribution,then
for
any
two
positive
integersm
and
q
prove
thatPIX>m+n/X>m]=PlX>nl.
ii.
Let
X
beis
a
normal
variatewith
mean30
and standard deviationis 5. Find
ttrefollowing
Pl26
<X
<
401,plx
>
4sland
P1:lx4oll>
5.
30.
a-
Two random samples gave the data.SamDle Size Mean Variance
I
2
8
li
9.616.5t)
2.t
Cao we conclude that the two samples have been drawn from the same normal population?
(oR)
b.i.
Test the significanceofthe
difference between the meansofthe
samples, dm\,,,rr from two normal populations with the same standarddeviationSamDle Size Mean SD
I
2 100 200 61 63 4 6ii.
The following data give the numberofair-craft
accidents that occured during the various daysof
a
week-Days
Mon
Tues WedTturs Fri
SalNo. ofaccidents
l5
19l3
t2
t6
15Test 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 numberof
customers that can be processed by the cashier is 24 per hour.(i)
whatis
the probability that the cashier is idle?(ii)
what is the averagenumber
of
customer in the queuing system?
(iii)
what is the aveEge time a customer spends in the system?(iv)
whal is the
avemge numberof
customersin
the
queue?(v)
whatis the
averagetime
a customer spends in ihe queue, waiting for service?(oR)
b.
The local one-person barber shop can accommodate a maximumof
5 people at a time (4 waiting andI
getting hair cut). Customers ardve according to a Poisson distribution with mean 5 per hour.The
baxber cutshair at an
avenlgemte
of
4
per hour
(exponential servicetime).
(i)
whatpercentage of time is the barber idle? (ii) what ftaction
ofthe
potential customers are tumed away?(iii)
what is the expected nunberof
customers waitingfor
a haircufl
(iv)
how much time can a customer expected to spend in tl-re barber shop?32.
a"
The transition probability matrixof
a Markov chain{\}:
n= t,
2,3...
having 3 states 1, 2, and 3fo.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 boysA,
B
andC
are throwing a ballto
each other.A
always thiows theball
toB
andB
always tl.[ows theball to
C, but C isjust
aslikety
totlrow
theball
toB
as toA.
ShowtlEt
theprocess 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]INGTMORY
(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 bookleLTime: Three Hours
1.
Max. Marks: 100
PART
-A
(20x 1=
20Marks)
AnswerALL
QuestionsThe probabiliry density
imction
ofa
contiruous random variableis
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:-
"-'-,
thenE(x)
is(B)
1/4(D)
1/6(B)
(D)
(c)
2.333.
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 moment1tt
(121=le,'1,,
-*.
r,.*
I
2Ja5.
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, thenE(-t
-t)'
represents6.
Ifthe
probability that a target is destroyed on any one shot is 0.5, then the probability that itwould
be destroyed on 6d attempt is(B)
0.0165(D)
0.0156(A)
0.106s(c)
0.r0567.
The cumulative distdbution fimctionoff(x)
ofan
exponential distribution is@)
"t'
-t;
x>o
@)
1-etu;
.r > o8.
The prcbability density function of a standard normal distribution is1
2l^(ts)
0e\=-=e
'
t'.0<z<*
\l
zlt
1
)/^
(D)
0lzl=-=e' t'i-a
< z<a
\lZJrrct
drzt
=Le'zl,g.
r.*
"l2z
10.
The test used to test the equaliry ofvariance
oftle
populations &om two small samples is(B)
F-test(D)
Independent of altributes20.
Statei
is said to be aperiodic with perioddi,
if
(A)
di>
1
(B)
d'<1
(c)
4=1
(D) di=o
(A)
t-test(C)
t'
.tests/ ',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) isal
,=P,
!
At
,=-!J:
slJn-l
.{
and population21NA4nA10t,l
&=gardan
0<x<l
elsewhere
ulnown
probabilitythe
steady statePART-B(5x4=20Marks)
Answer
ANY FIVE
Questio.sA
random variableX
hasa
meanp
=
12 and variance distribulion. Find P(6<X<l 8).22.
23.
25.
26. 21.
Given the random variable
X
with density fimction Find thepobability
densiryofY
= 8X3.A
favel
company hastwo
carsfor hiring.
The demandfor
a cztl on each dayis
distributed asPoisson variate with mean 1.5. Calculate the
propotion
ofdays on which some demand is refused'24,
Thetime (in hous) requied to
repair
a
maohineis
exponentially distributedwith
parameterL
=