andis equalto
0 otherwise.The
Problems 247
constantB(a,
,) is givenby
tl
B(a, b.)
x"-t1l - ry-t
'fhe
ot
Jo
meanandvariance
ofsuc nu=
random
Etwt
't^t='
4
;;i
PRoBtEMs
variableare
Var{xr=
.-E--.-.-.__\-,u- rD
) ' t ar b + t )
l. Let X be a
randomyariabie
witl l probability
densityfuncdon
Irxt=
1a) w,hatis thevalue
ofc.,
lc{l
--r2) -l
<x.l
otherwise
,. ;
runcrion
orx?
otL
me
".,l'J1;.,$,TTi,jjl:^9::,11,,'on
x tt't'"0.""'ii *' ii
" "i J,lJii;tln:jiil:f,il,tl"." rorarandom
amounr
fzt = Icrc-'/2 x >-o
that
the
sysrem
,. J#i,'.'":Tflix:tJ:'jr
,"",.", ..
monrhs?
",i".0,,5
I r x t = l c r 2 x - L c 3 1g ' x - !
;::llf
"
aprobabiiitv
a"n,rrv
rl,"n",ronr
,r.".l',i,". .o.
"{"n'
f al = lctz' . '2) 0 *' '
"
othe.ruire2
if /{") t"t"
4' Theprobabiiiry
d.nri,u0,.^,,^- ^ll
ofx, therifetime
trneasu.eo
rnrou.st,;;;*;tT
ofa cerrainrype
ofejecrronic
device
tat={# .-to
t a t F i n dp l x - ) ^ t
lo
r 5 lo
lli ,yffli f:;,TlH;t;,i,:l:r'*
orx?
nncrion
h'.".,
s. An*:::1^1: "r',i'i"",'ll,l"'JL'il:';ff;J;."r reasr
r wirrruncrion
fora,
d;;"..#g;;'11;'";i*'"il"[i"*lr#],lil
voi,,
me
orvh.;n
nnk,]
'
fr'r= /s't -xf o'x'l
orher*ise
or rh"runkoe
',.u.rcu
liil#:r:i:
"apacirr
,,
so
that the
u a given
weekis .0l?
probabiriryof
the suppry,rbeing
ex-
248 chapter 5
continuousRandomVariables
6. ComputeE[X] if X hasa densityfunctiongivenby
l!'"'/2 ^'o
(a, /(r):14
[o
:
o*rerwrse
lc(l-.^2) - l<,r< I.
olherwise
l0
f5
.r>)
l;
(c) /(r) :
.{'
|
- r: 5
[o
7. The densityfunctionof X is givenby
lD) JI(r-
:r:l
f @:16*b',ootherwise
If EtXl = l, 6nd a andD.
probability
8. The tifetimi in hours of an electonic tube is a rundom variable having a
density
' functiongivenbY
r-0
f Qc)--xe-'
Computethe expectedlifetime of such a tube.
is a con9. ConsiderExample4b of Chapter4, but now supposethat the seasoraldemand
the
optimal
that
Show
function
/
tinuous random variable having probability density
satisf,es
amountto stock is the value s* that
F6\ =:-
t) f (
the cumulatrve
where, is netpront per unit sale,I is thenet lossper unit unsold,andF is
demand'
seasonal
of
the
function
distribution
intervals starting
10. Trains headedfor destination A arrive at the hain station at 15-minute
intervalsstarting
l5-minute
at
at 7 A.M., whercasbains headedfor destinationB arrive
at 7:05A.M.
(a) If a certain passengerarrives at the station at a time unifonnly distributed between
of time
7 and 8 e.v. and then gets on the first train that ardves, what proportion
doeshe or shego to destinationA?
(b) Whatif the pasiengerarrivesat a time uniformly distributedbetween7: l0 and8:l0
A.M.?
statementand
11. A point is chosenat randomon a line segmentof length t lnterpret.this
less
than i
is
segment
;nger
the
to
sfiorter
oiihe
ratro
that
ttre
probability
find the
.
12.AbustravelsbetweenthetwocitiesAandB,whicharel00milesapart.Ifthebushasa
over (0'
breakdown,the distanceftom the breakdownto city A has a uniform distribution
route
between
of
the
center
in
the
B'
and
A,
in
city
100).Thereis a busseNicestationin
that it would be moreeffrcientto havethe threestationslocated
A and B. It is suggested
ftom A' Do you agree?Why?
respectively,
milis,
75
25, 50, and
at sometlme
13. You arrive at a bus stop at 10 o'clock' knowing that the bus will arive
uniformly distributedbetweenl0 and 10:30'
(a) What is the probabilitythat you will haveto wait longerthan l0 minutes?
(b) If at l0:15 the bushasnot yet airived,what is the prcbabilitythat you will haveto
wait at leastan additionall0 minutes?
|4,LetXbgauniform(0,].)randomvariable.comPuteE[x,]byusingPloposition2.l
andrhenchecl theresultb1 usingthedefinitionof expectation'
Problems 249
a probability
hecumulative
rvals starting
rvals starting
rutedbetween
ortionof time
7 : 1 0a n d8 : 1 0
statementand
s than i.
fthe bushasa
[tion over(0.
oute between
ltions located
at sometime
utes?
u will haveto
Dposition2.1
15. If X is a normalmndomvariablewith parameters =
1,r 10 ando2 : 36, compute
(a) P{X > 5}:
(b) P{4<X<16}:
(c) P{x < 8};
(d) P(x < 201;
(e) P{X > 16).
16. The amual rainfall (in inches) in a certain region is normally
distributed with lt, : 40
and q : 4. What is rhe probability that starting with thi, y""r,
it *ili tut" over 10 years
before a year occurshaving a rainfall of ovei 50 ln"nesf
Wtat urru_puon" u." you
making?
17. A man aiming at a target receives 10 points if his shot is
within I inch of the target, 5
pointsif it is between1 and 3 inchesof the target,and points
3
if it is between3 and 5
inchesof the target.Find the expectednumberof points scored
if the distancefrom the
shot to the target is uniformly distributed betweenb and 10.
18. Supposethat X is a normalrandomvariablewith rnean5. p{X
If
> g} = .2, approxi_
matelywhat is Var(X)?
19. L€t X be a normal randomva.iablewith mean 12 and variance
4. Find the value of c
suchthat PIX > c] = .10.
20. If.65 percent of the population of a large conmunity is in
favor of a proposed rise in
schooltaxes,approximatethe Fobability that a randomsanple
of I00 peoplewill con_
tarn
(a) at least50 who are in favorof the proposition;
(b) between60 and 70 inclusive who are in favor:
tc) fewer rhan75 in favor
21. supposethattheheight,in_inches,
of a 25-year-oldmanis a normalrundomvariabrewith
paxametersg. : '1I ando2 = 6.25. What percentageof 25_year_otJ
men are over 6 feet
2 inchestall? what percentage
of men in the 6-fooierclub a.e ou". e ioot s inctr"sr
-22. The width of a slot of a dumluminforgingis (in inches)
normallydistriOuteawith pr.=
.9000ando : .0030.The specificationlimits weregivenas .SOOb
j OOSO.
(a) What percentageof forgings will be defective?
(b) What is the maximumallowablevalueof o that will permit
no more thar 1 in 100
defectivesWhenthe widthsare normallydistributedwith p = .9000
ando?
23. One thousandindependentrolls of.a fair die will be made,
Compur" an upproxlmatron
to the probabilitythat number6 will appea{berween150 and 20b
dmesrnclusively.If
number6 appearsexactly200 times,find the probabilitythat
number5 will appearless
than 150times.
24. The lifetimesof interactivecomputerchips producedby a
certainsemlconductor
man_
are n-ormally distributed with parametersp = 1.4 x. 106hours
and o :
:rac,r:le:
r x I uwhat rstle approximateprobabilitythat a batchof 100chipswill
-n-ouls.
contain
at least20 whoselifetimesare lessthanl.g x 106i
25. Each item produced by a certain manufacturer is, independently,
of acceptablequality
with probabiiiry.g5. Apprcximatethe probab ity that at most iri qith"
ne*t tso itern"
produced arc unacceptable.
26. Two_types of coins are produced at a factory: a fair coin and
a biasedone that comesup
heads55 percentof the time. We haveone of thesecoinsbut do
noifiow wtretherit is
a fail coin or a biasedone. In orderto ascertainwhich type of coin
we have,we shall
peform the following statisticartest:we sharltossthe coin
1000times.rfthe coi,' lands
on heads525 or more times,thenwe shall concludethat it is a
biasedcoin, whereas,if
it landsheadslessthan 525 times,then we shall concludethat it is
the tair coin. Ifthe
250 Chapter5
continuousRandomVariables
coin is actuallyfair, what is the probabilitythat we shallreacha falseconclusion?What
would it be if the coin were biased?
tossesof a coin,the coin landedheads5800times ls it reasonable
27. In 10,000independent
to assumethat the coin is IIot fair? Explain.
28. Twelve percent of the population is lefthanded. Approximate the probability that there
are at least20 lefthandorsin a schoolof200 students.Stateyour assumptions.
29. A model for the movementof a stock supposesthat if the presentpdce of the stock is s.
then after one time pedod it wilt be eitherus with Fobability p, or ds with Fobability I
p. Assuming that successivemovementsarc independent,apForimate the probability
that the stock'sprice will be up at least30 percentafter the next 1000time periodsif
u : 1.0\2, d : 0.990,andp : .52.
30. An image is partitioned into 2 regions--one white and the other black. A rcading taken
from a mndomly chosenpoint in the white section will give a reading that is nomally
distributed with trr : 4 ando2 : 4, whereasone taken from a mndomly chosenpoint in
the black region will have a normally distributed reading with parameters(6, 9). A point
is randomly chosenon the image and has a reading of 5. If the fraction of the image that
is black is o, for what value of d would the Fobability of making an eflor be the same
whether one concludedthe point was in the black region or in the white region?
31. (a) A fire stationis to be locatedalonga road of lengthA, A < a. If fires will occur
at points uniformly chosenon (0, A), where should the station be located so as to
rninimize the expecteddistanceftom the nre? That is, choosea so as to
minimizeEIX - all
whenX is uniformly distributedover (0, A).
(b) Now supposethat the road is of infinite length-stetching ftom point 0 outward
to oo. If the distance of a fire from point 0 is exponeltially distributed with rate
;", where should the fire station now be located? That is, we want to minimize
E[lX - al], whereX is now exponentialwith rate.]".
32. The time (in hoursI requiredto repaira rnachineis an exponentiallydistributedrandom
I : {. What is
variablewith Darameter
(a) the probability that a rcpair time exceeds2 hou$;
(b) the conditional probability that a rePair takes at least 10 hours, given that its dulation exceedst hours?
33, The number of years a radio functions is exponentially distributed with parameter)" :
probability that it will be working after an
{. If Jonesbuys a usedradio, what is the
additional8 years?
34. Jonesfigures that the total number of thousandsof miles that an auto can be driven beforc
it would needto be junked is an exponential rundom variable with parameter S Smith
has a usedcar that he clairnshasbeendriven only 10,000miles. If Jonespurchasesthe
car, what is the probability that she would get at least 20,000 additional miles out of
it? Repeatunder the assumptionthat the lifetime mileage of the car is not exponentially
distdbuted but nther is (in thousandsof miles) uniformly distributed over (0, 40).
35. The lung cancerhazardmte ofa /-year-oldmale smoker,l(l), is suchthat
^\t):.o27 -.00025u 4ot2
tz4o
Assuming that a 4O-year-oldmale smoker survives all other hazards,what is the probabitty that he survives to (a) age 50 and (b) age 60 without contacting lung cancer?
36. Supposethat the life distributionof an item hashazardmte function)'(/) : t3,t > O.
What is the probability that
Problems' 313
'E b
v
wherep is theirjoint probabilitymassfunction.Also, if X andf arejointly
continuous
with joint d€nsityfunction /, then the conditionalprobability den"sity
function of X
giventhat I = y is givenby
IxvQli:
f \:4
f v \.Y)
vafuesX111.5Xot 1
X1,1of a setof independent
andidenticay
l!::L*:o
,,.. !
orstnbutedrandomvariablesare called
the order statisticsof that set. If the random
variablesarecontinuouswith densityfunction thenthejoint densityfunction
/,
of the
orderstatisticsis
J@t"
' x " ) : n t f ( x r )" ' f ( x , )
xt=x2=..'=xh
The randomvariablesX1, . . . , Xn are calledexchangeable
if thejoint distributionof
X ir, . . ., X i, is the samefor everypermutation
ir, . . ., i,, of 1, . . ., n _
PROBLEMS
1. Twofairdicearerolled.Findthejoint probabilitymassfunctionof X andf when
(a) X is thelargestvalueobtained
on anydieandy is thesumof thevatues:
(b) X is thevalueonthefimtdieandy is thelargerof thetwo values;
(c) X is thesmallest
andy is thelargestvalueobtained
onthedice.
2. Suppose
that3 ballsarechosenwithoutreplacement
from anum conslstrng
of 5 white
and8 redballs.LetX; equalI if thelth ballselected
is white,andletit equal0 otherwise.
Givethejoint prcbabiliry
masstunctionof
(a) X1,X2;
(b) Xt, Xz, Xy
3. In-Problem2, suppose
thatthe whireballsarenumbered,
andler fi equalI if thelth
whiteballis selected
and0 otherwise.
Findrhejointprobability
massfunctionof
(a) Yt, Y2:
(b) Yt, Yz,Yz.
4. RepeatProblem2 whenthe ban selectedis replacedin the um beforcthe next
serection.
5. RepeatProblem3a whentheba[ selectedis replacedin the um beforethe next
selection.
6. A bin of5 tansistorsis known to contain2 that are defective.The transistors
are to be
tested,oneat a time, until the defectiveonesareidentified.Denoteby N1 the number
of
testsmadeuntil the first defectiveis identifledand by N2 the numbeiof additional
tests
until the seconddefectiveis identified;find thejoint probabilitymassfunctionof
Nr and
7. Considera sequenceof independentBemoulli trials, each of which is a success
with
probabilityp. Let Xl be the numberof failuresprecedingthe flrct success,
and tet X2be
the numberof failuresbetweenthe fiIst two successes.
Find the joint massfunctionof
Xl andX2,
8. Thejoint probabilitydensityfunctionof X and y is sivenbv
f ( x , y ) : c 1 y 2- y z p - t
(a) Find c.
(b) FindLhemarginal
densiries
of X and f.
(c.) Find E[X].
-)<xs),0<-y<oo
314 Chapter6
JointlyDistributedRandomVariables
9. The joint probability density function of X and y is given by
. r ( . r , v i)Q
: '.+)
o < x <r ,o . v . 2
(a) Verify that this is indeedajoint densityfunction.
(b) Computethe densityfunctionof X.
(c) Find P{X > YJ.
(d) FindP{r r;X.;1.
(e) Find E[x].
(f) Find E[Y].
10. The joint probabilitydensityfunctionof X and f is givenby
ir")
/ ( . r . t y )= e
0.x<oo.0.y<oo
Find(a) P{X < Y} and(b) PIX < al.
11. A televisionstoreowner figurcsthat 45 percentofthe customersenteringhis storewill
purchasean ordinarytelevisionset, 15percentwill purchasea plasmatelevisionset,and
40 percentwill just be browsing.If 5 custome$enterhis storc on a given day,what is
the probabilitythathe will sell exactly2 ordinarysetsand 1 plasmaseton that day?
12. The number of people that enter a drugstorein a given hour i$a Poissonrudom vadable
with pammeter)" : 10. Computethe conditionalprobabilitythat at most 3 men entered
the drugstore,given that 10 women enteredin that hour. What assumptionshaveyou
made?
13. A man and a woman agreeto meet at a certainlocation about 12:30P.M. If the man
arrivesat a time uniformly distributedbetween12:15 and 72.45 and if the woman independentlyarrivesat a time uniformly distributedbetween12:00and I PM., lind the
Fobability that the flISt to arrive waits no longer than 5 minutes. What is the probability
that the man anives flIst?
L4. An ambulancetravelsbackand forth, at a constantspeed,alonga rcad of lengthl. At a
certainmomentof time an accidentoccursat a point uniformly distributedon the road.
[That is, its distancefrom one of the fixed ends of the road is uniformly distributed over
(0, t).1 Assumingthat the ambulance'slocation at the momentof the accidentis also
uniformly dist buted,compute,assumingindependence,
the distributionofits distance
ftom the accident.
15. The randomvector(X, y) is saidto be unifondy distributedovera regionR in the plane
if, for someconstantc, itsjoint densityis
- c
r, ,' ', r, ',-, ll 0
i f t . r . Y re f i
otherwise
(a) Showthat 1/c: areaofregion R.
Supposethat (X, Y) is unifornly distributedover the squarecenteredat (0, 0), whose
sidesarc oflength 2.
(b) Show that X and f are independent,with eachbeing distdbutedunifbrmly over
(-1. 1).
(c) What is the probability that (X, y) lies in the circle of mdius 1 centeredat the
origin?That is, find P{X2 + Y2 < \1.
16, Supposethat r points are independentlychosenat mndomon the perimeterof a circle.
and we want the probabilitythat they all lie in somesemicircle.(That is, we want the
probabilitythat therc is a line passingthroughthe centerof the circle suchthat all the
pointsareon one sideof that line.)
Problems315
Let Pt,. . ., Pz denotethe,' points. Let A denotethe eventthat all the points are contained in some semjcircle, and let A; be the event that all the points lie in the semicircle
beginningat the point P; andgoing clockwisefor 180', i : l, . . ,n.
(at ExpressA in lermsofthe Ai.
(b) Are the A; mutually exclusive?
(c) Find P(A).
17. Threepoints X l, X2, X3 are selectedat randomon a line Z. Wlat is the probabilitythat
X2 lies betweenXt and X3?
lE. Two points a{e selectedrandoniy on a line of length I so as to be on opposite sides
of the midpoirt of the line, [In other words, the two points X and Y arc independent
random variables such that X is uniformly distributed over (0, Ll2) and Y is uniformly
distributed over (L/2, Z).1 Find the probability that the distancebetweenthe two points
is geater than,L/3.
19. Showthat /(t, y) : l /x,0 < y < )c < I is ajoint densityfunction. Assumingthat /
is thejoint densityfunctionof X, 1, find
(a) the marginal density of Y;
(b) the maryinal density of X;
(c) E[x]:
(d) Etvl.
20. The joint demity of X and y is givenby
/(,, r) : {;'-''*"
Are X and ts independent?What if /(t,
,rtx. ),=
.r>0, y>0
otherwise
y) were given by
12 0<x<y.0<)<
l0 otherwise
L
21. Let
f6,y):24xy
o5t:1'
o5y:1'
o5x*Yl1
and 1etit equal 0 otherwise.
(a) Showthat /(r, )) is ajoint probabilitydensityfunction.
(b) Find Elxl.
(c) Find ElYl.
22. The joirt density function of x and y is
y < I
: l,+v 0 < x < l, 0 <
"f(,r,r) l 0
otherwise
316 Chapter6
JointlyDistributedRandomVariables
(a) Are X and Y independent?
(b) Find the densityfunctionof X.
(c)FindPIX+Y < 1].
23. The randomvariablesX and I havejoint densityfunction.
-f(r,y):12ry(1
"t)
0<r<1,
0<)<l
and equalto 0 otherwise.
(a) Are X and I/ independent?
(b) Find E[x].
(c) Find E[Y].
(d) Find Var(x).
(e) Find Var(Y).
24. Considerindependenttrials each of which resultsin outcomei, i :
0, l, ..., i with
t
25,
26.
27.
28.
29.
probabilityh , D p; ,:' | . Let ly' denotethe numberof trials neededto obtain an outi:0
comethat is not equalto 0, andlet X be that outcome.
(a) FindP{N:nl,nZl.
(b) FindP{x: j), j =r,...,k.
( c ) S h o wt h a tP { N : n , X = j \ = P { N : n \ P l X : j J .
(d) Is it intuitiveto you that N is independent
of X?
(e) Is it intuitiveto you that X is independent
of N?
Supposethat 106peopleanive at a servicestationat timesthat areindependent
mndom
variabl€s,eachof which is uniformly distributedover (0, 106).Let ly' denotethe number
that arive in the nrst hour.Find an approximationfor P{N : il.
Supposethat A, B, C, arc independentrandom variables,each being uniformly distdbutedover (0, l).
(a) What is thejoint cumulativedistuibutionfunctionof .A,B, C?
(b) What is the probabilitythat all of the rootsofthe equationAx2 + Bx + C : 0 are
real?
If X is uniformly distributedover(0, l) and Y is exponentiallydistributedwith parameter
),: l, findthedistribution
of(a) Z: X+Y afi (b) Z: X/l'. AssumeindependenceIf X1 andX2 are independentexponentialnndom vadableswith respectiveparameren
11 and.l"2,find the distributionof Z : Xt/Xz. Also computeP{X1 < X2l.
When a current 1 (measuredin amperes)flows through a rcsistanceR (measuredin
ohms),the power generatedis givenby Iry': 12R (measuredin watts). Supposethat /
and R are independent
rundomvariableswith densities
-6.111
fi(.r)
.fp(.:t)= 2x
\)
0S,r<1
0<x<l
Determine the density function of l/.
30. The expectednumberoftypogmphicalerrorson a pageofa certainmagazineis .2. What
is theprobabilitythatanarticleof 10pagescontains(a) 0, and(b) 2 or moretypog.aphical
errors?Explainyour reasoning!
31. The monthly worldwide avercgenumberof airplanecrashesof comnercial airlinesis
2.2. What is the probabilitythat therewill be
(a) morethan2 suchaccidentsin the next month;
(b) morethan4 suchaccidentsin the next 2 months;
Problems 317
(c) more than 5 such accidentsin the next 3 months?
Explail youl reasoning
!
The gross weekly salesat a certain restaurantis a normal random vadable with mean
$2200 and standarddeviation $230. What is the prcbability that
(a) the total gross salesover the next 2 weeks exceeds$5000;
(b) weekly salesexceed$2000 in at least 2 of the next 3 weeks?
Wlat independenceassumptionshave you made?
.'-t. Jill's bowling scoresare approximately normally distributed with mean 170 and standard
deviation 20, while Jack's scoresaxeapproximately normally distributed with mean 160
and standarddeviation 15. If Jack and Jill eachbowl one game,then assumingthat their
scoresare independentrandom variables, apprcximate the probability that
(a) Jack'sscoreis higher; ''
(b) the total of their scoresis above350.
34. Accordingto the U.S. NationalCenterfor Health Statistics,25.2 percentof malgsand
23.6 percent of females never eat breakfast. Supposethat nndom samplesof 200 meII
and 200 women arebhosen.Approximate the probability that
(a) at least110of these400 peoplenevereatbreakfasti
(b) the number of the women who never eat breakfastis at least as large asthe number
of the men who never eat breakfast.
35. In Problem 2, calculate the conditional probability massfunction of X1 given that
(a) Xz = 1;
(b) x2 = 0.
36. In Problem 4, calculate the conditional probability massfunction of Xt given that
(a) X2 = l;
(b) x2 : 0.
37. In Problem 3, calculatethe conditional probability massfunction of yl giYenthat
(a) Y2: ll
(b) v2:0.
38. In Problem 5, calculate the conditional probability massfunction of yl given that
(7) Y2=l;
(b) v2:0.
39. Choosea number X at randomfrom the set of numbers{1,2,3,4,51. Now choosea
numberat randomfrom the subsetno largerthan X, that is, from {1,..., X}. Call this
secondnumberI.
(a) Find the joint massfunction of X and y.
(t) Findtheconditional
massfunctionofx giventhatf = i. Doitfori :1,2,3,4,5.
(c) Are X and l independent?Why?
40. Two dice are rolled. Let X and Y denote, respectively,the largest and smallest values
obtained.Cornputethe conditionalmassfunctionof I givenX : i, for i : 1,2, .. .,6.
Are X and Y independent?Why?
4L. The joint Fobability massfunction of X and y is given by
p(l,l): *
P ( r , 2 ) :i
p ( 2 , \ ) :i
p Q , 2 ) :+
(a) Computethe conditionalmassfunctionof X given Y : i,i : 1,2.
(b) Are X and Y independent?
(c)compute
P{xv <31,Plx +Y >21,PlxlY > ll.
3 1 8 C h a p t e6r
JointlyDistributedRandomVariables
42. The joint densityfunctionof X and Y is givenby
f(x,y):xe
r()+r) ;r>0,
),>o
(a) Find the conditionaldensityof X, given f : )], andthat of I, givenX : .r.
(b) Find the densityfunctionof Z = Xy.
43. Thejoint densityof X and Y is
f ( : r ,D : c ( x 2- ! 2 ) e - '
o<ir<c!,
-r:y:.r
Find the conditionaldistributionof Y, given X : -r.
44, An insurancecompanysupposesthat eachpersonhas an accidentpammeterand that
the yearly nurnberof accidentsof someonewhose accidentparameteris .1"is poisson
dist butedwith mean),. They alsosupposethat the pammetervalueof a newly insured
personcan be assumedto be the valueof a gammarandomvariablewith pammeters.t
and a. If a newly insuredpersonhas n accidentsin her flrst year,find the conditional
densityofher accidentparameterAlso, determinethe expectednumberofaccidentsthat
shewill havein the following year.
45. lf X1, X2, X3 are independentrandomvadablesthat are uniformly distributedover (0.
l), computethe probability that the largestof the three is greaterthan the sum of the
othertwo.
46. A complexmachineis ableto operateeffectivelyaslong as at least3 ofits 5 motorsare
functioning. If eachmotor independentlyfunctionsfor a randomamountof time wirh
densityfunction/(r) = xe-' , x > 0, computethe densityfunctionofthe lengthof time
that the machinefulctions.
47. If 3 fucks break down at points randomlydistriburedon a roadtf length t, iind the
probabilitythat no 2 of the tucks arc wirhin a distanced of eachotherwhend : L/2.
48. Considera sampleof size5 from a uniform distributionover (0, 1). Computethe probability that the medianis in the interval ( {, ol).
49. If Xt. X2, X3, X4, X5 are independentand identically distributedexponentialrandom
variables with the parameter)", compute
(a) P{min(Xr,...,X5) : rr};
( b ) P [ m a x ( X 1 , . . .X, 5 ) : a ] .
50. Derive the distributionof the range of a sampleof size 2 from a distributionhaving
densityfunction
/(r) = 2r,0 < r < 1.
51. Let X and y denotethe coordinatesof a point unifondy chosenin the circle of radiusI
centeredat the origin. That is, theirjoint densityis
I
x2+!2=1
7f
Find_thejoint densityfunction of the polar coodinates R : (X2 + y\1/2 and@ an tYlx.
52. If X and I areindependentrandomvariablesboth uniformly distributedover (0, l), find
t h e j o i n dr e n s i t fyu n c r i o on f R - , , - x , - Y ' - O - - t a n - t y / x .
{ ? If U is uniform on (0, 2z) and Z, independentof U, is exponentialwith rate 1,
show
dircctly (withoutusingthe resultsof Example7b) thar X and Y definedby
X : J22co:u
Y : xEZ sinU
areindependent
standardnormalrandomvariables.
Theoretical Exercises 319
54. X and I havejoint densityfunchon
ytr. tt:
9ter and that
I is Poisson
twly insured
Dalamelen.r
) conditional
rcidentsthar
ftedover (0.
I sum of the
i motors are
f time with
ngthof time
l,, find the
d < L/2.
e the probatial random
fon having
ofradiusI
(0, l), find
Cel, shou
)zrz
.r> r. ),: r
(a) Computerhejoint deNity functionof U =
Xy,V = X/y.
(b) Whar arc the marginal densities?
55. If X arld f are independentand identically distributed
uniform random variables on (0,
l), computethejoint densityof
(a) U=X+y,V:X/y;
(b) U : X,V : X/Y;
( c )U = X + Y , V : X / ( x + y ) .
56. RepeatProblem 55 when X and I are independentexponential
mndom variables, each
with parameter), = l.
57, tf X1.andX" are independent
exponential
randomvariables
each
"""' havrngparameter
).,
findthejointdersig,functionof I/1: Xt!Xzandyz:ei,.
_58. If,X, y, and arc independent_random
variableshaving identical density functions
-Z
e', 0 < r < oo, derivethejoint distributionof U": i
i i, v = X + z, w =
dfi) ;
59. In Example8b,1etY1a1
: na1.
fy,.
Showtharyr..... y*,yr+tare exchangeable.
Note that fft+l is the number of billi one must observe
to obtain a special ball if one
consldersthe ballsin their reverseorderof withdrawal.
60. Consideran.urncontaining/, balls,numberedl, .. ., z,
and supposethat f of them are
randomllwirhdrawn.Ler Xi equalI if ball numbered
i i, re.o'u!aunOlerir be 0 orher_
w l s e .) n o w l h a lX t . . . . . & a { ee x c h a n s e a b l e .
THEORETICAL
EXERCTSES
1. VeriryEquation
(1.2).
2. Supposethat the numberof eventsthat occurin a given
time periodis a poissorrandom
variable with parameter,,. If each event is classified as a type
I event with probability
pi,i = 1,...,n,Dp;
= 1, independently
of otherevents,showthatthe numbersof
t)4'e i eventsthat occur,i : 1,...,n, are independentpoisson
mndom variableswith
respecnveparameteN
),pi,i : 1,..., n,
3. Suggesta procedurefor using Buffon,s needleproblem to
estimatez. Suryrisingly
enough,this wasoncea commonmethodof evaluatinsz.
4. SolveBuffon s needleproblemwhenI > D.
a N s w e n :1 1 ( l - s in e | + 2 e/ n . w h ere d i s s u c hr h a cr o s
0: D/ L.
5. If X and f are independentcontinuous positive random
variables, expressthe density
functior of (a) z = X /Y afi (b) z = Xr in termsof the a"nrity
iun"tion, or x una
y. Evaluatetheseexpressions
in the specialcasewhere X and y"ale mtn exponential
random va.riables.
6. Showanalytically(by induction)that Xr * . . . * Xn hasa negative
binomiatdistribution
whenthe X;, I : 7, . . -, n a.reindependent
artdidenticattydiitributeJgeornetricrandom
variables..Also, give a secold argumentthat yedfies the
;receding wiitout any need tor
computahons.
7. (a) If X has a gammadistributionwith paramete$(r,.i.),
what is the distributionof
cX,c>0?