Download ity density function that characterizes the proportion Y that make a

results by means of a
(b) Evaluate F (x).
(c) What is the probability that a random particle
3.31 from
Based
on extensivefuel
testing,
determined by
the manufactured
exceedsit4 is
micrometers?
3.34 Magnetron
4 dimes and 2 nickels, the manufacturer of a washing machine that the time assembly line. A
without replacement. Y 3.30
(in years)
before aofmajor
repair
is required
is char- assess quality of
Measurements
scientific
systems
are always
n for the totalEX
T 1:
of Measurements
the subject of
variation,
someare
more
than
others.
There some more
scientific
systems
always
subject
to variation,
than
surement
is subj
acterized to
by
the probability
density
function
ty distribution
graphiaremany
manystructures
structures
for
measurement
error,
and
statisothers.
There are
for
measurement
error,
and
statisticians
spend
a
great
deal
of
probability
t
! 1 −y/4
94
Chapter
3 the
Random
Variab
.
spendSuppose
a greatthe
deal
of time modeling
these
errors.
time modelingticians
these errors.
measurement
error X of
a certain
physical quantity is
e
,
y ≥ 0,
ification is 0.99.
f (y)
= 4
the
measurement
error X of a certain physical
decided by theSuppose
density function
elsewhere.
lengths of 5 rand
black balls and 2 green quantity is decided 0,
by the density
function
ession, each ball being 3.31 Based on
Magnetron
testing, it is determined by 3.34
! extensive
(a) Show
that tht
2
k(3
−
x
),
−1
≤
x
≤
1,
ext draw is made. Find (a)the
Critics
would
certainly
consider
the
product
a
barline.ofA5 sa
manufacturer
of a washing machine that the time assembly
f (x) =
ber out
t
r the number of green Y gain
0, a major
elsewhere.
if it is before
unlikely
to require
a major
repair
(in years)
repair
is required
is before
char- assess quality of th
by the follow
sixth
Comment
on
this
by determining
surement is subjec
acterized
by year.
the
probability
density
function
(a)the
Determine
k that
renders f (x)
a valid
density
funca) Determine
k(Y
that>renders
density function.
Sol:the
3/16probability tha
Ption.
6). f(x)!a valid
hours of an important
1 −y/4
e
,
y
≥
0,
b)
Find
the
probability
that
a
random
error
in
measurement
is
less
than
1/2.
Sol:
99/128 is f
(y) =
ification
0.99.
A
4
sed in a manufactured(b)
isfthe
probability
a major
occurs
(b)What
Find the
probability
that that
a random
errorrepair
in
mea(y)
=
94
Chapter
3
Random
Variables and P
c)
For
this
particular
measurement,
it
is
undesirable
if
the
magnitude
of
the
error
(i.e.,
|x|)
0, 1/2. elsewhere.
lengths of 5 random
nction
is less than
insurement
the
exceeds 0.8.
Whatfirst
is theyear?
probability that this occurs?
Sol: 0.164
(a) Show
(c) For this particular measurement, it is undesirable
for ythat
= 0,the
1, 2p
00), x ≥ 0,
(a) 3.31
Critics
would
certainly
consider
product
3.34 Magnetron
tubes
are
Based
on extensive
testing,
it isthe
determined
bya barif
the
magnitude
of
the
error
(i.e.,
|x|)
exceeds
0.8.
ber
out
of
5
tha
EX
2:
Based
on
extensive
testing,
it
is
determined
by
the
manufacturer
of
a
washing
machine
3.32 the
The
budget
for
certain
type
assembly
line.
A sampling
p
manufacturer
of aof
washing
machine
thatarepair
the
time
x < 0.
gain
ifisproportion
itthe
is unlikely
tothe
require
aoccurs?
major
before
(b)
Suppose
rand
What
probability
that
this
by of
thethefollowin
that the time
(inYyears)
before
a major
repairisisrepair
required
isto
characterized
by assess
the probability
quality
lengths
(in
years)
before
a
major
is
required
is
charofYindustrial
company
that
allotted
environmental
the sixth year. Comment on this by determining
3 are
out
density function
surement is and
subject
to uncer
acterized bycontrol
the probability
density
function
and pollution
is coming
under
scrutiny. A data
P (Y > 6).
ther
toasuppo
the probability
that
random
! 1 −y/4
collection
project
determines
that
the
distribution
of
e
,
y
≥
0,
f
(y)
= is
ification
is
0.99.
A
sampling
4
probability
(b) What isfthe
probability that a major repair occurs
(y) =
y!
these in
proportions
is
given
by
0,
elsewhere.
lengths
of
5
random
tubes
a
the first year?
cations.
(a) Show that the probabilit
! certainly
a) Critics would
consider
the product
a bargain
if it is unlikely
4consider
(a) certainly
Critics would
the
product
a bar- to require a major
= 0,meet
1, 2,len
3
5(1
−
y)
,
0
≤
y
≤
1,
ber 0.2231
outfor
of 5ythat
repair before the
year.
Comment
on this by
determining
P(Y
>6).
ifproportion
it=
is unlikely
to require
a major
before
fsixth
(y)
3.35
Suppose
it
3.32 gain
The
of
the
budget
forrepair
a certain
typeSol:
by (b)
the
following
Supposediscrete
rando
b) What is the probability
a major
repair occurs
in the
first year?
Sol:
0.2212
0,
elsewhere.
the sixththat
year.
Comment
on
this
by determining
torical
data
that
of industrial
company
that
is
allotted
to
environmental
and 3 are5!outsi
P (Y > 6).
a
specific
interse
and
pollution
control
coming
under
scrutiny.
A data
f (y)
(0
EX 3: The (a)
proportion
of that
the is
budget
for
aiscertain
industrial
that is allotted
to= to support
ther
Verify
theprobability
above
isthat
atype
valid
density
function.
(b) What
the
a of
major
repaircompany
occurs
y!(5 − y)!
collection
project
determines
that
the
distribution
of
is
characterized
environmental and pollution
control
under scrutiny. A data collection project
in the
first
year?is comingthat
probability is 0
(b)
What
is
the
probability
a company chosen at function:
these
proportions
is
given
by
determines that the distribution of these proportions is given by
0, 1, 2, 3, 4, 5.
random expends
less than 10% of its budget on en-for y =cations.
!
3.32 The proportion of the
budget for a certain type (b) Suppose random selectio
4
vironmental
and
pollution
5(1
−
y)is, allotted
0controls?
≤ yto≤environmental
1,
of industrial company
that
and3.35
3 are outside
specific
f
(y)
=
Suppose
f (x)
=itei−
and pollution
under
A data
0, is comingthat
elsewhere.
(c) What
is the control
probability
ascrutiny.
company
selectedthertorical
to support
to refu
dataorthat
X
collection
project
determines
that 50%
the distribution
of
probability is 0.99 that a
at
random
spends
more
than
of
its
budget
on
a) Verify (a)
thatthese
the above
is a valid
density
function.
Sol: ? cations.
a specific intersect
proportions
isabove
given by
Verify
that
theand
is a valid
density function.
Find the pro
environmental
pollution
controls?
b) What is the
probability that !
a company
chosen
at
random
expends
less
than
10%
of its
is(a)
characterized
by
4
5(1
− y) controls?
, that
0 ≤ y a≤company
1,
time
period,
What isf (y)
the=
probability
chosen
at function:
budget(b)
on environmental
and
pollution
Sol:3.35
0.4095
Suppose it is known f
0, less than
elsewhere.
expends
10%
of itsdata
budget
on enintersection.
c) What
is therandom
probability
that
a company
selected
at random
spends
more
than 50%
of its
torical
data
that X, the num
3.33
Suppose
a certain
type
of
small
processing
vironmental
and
pollution
controls?
budget
on
environmental
and
pollution
controls?
Sol:
0.031245
a
specific
intersection
during
the
pro
−6
firm is
specialized
that issome
difficulty
(a)so
Verify
that the above
a validhave
density
function.making (b) Find
fby
(x)the
= efollo
is
characterized
(c)
What
is
the
probability
that
a
company
selected
a profit
in type
their
first
year
ofthat
operation.
probabilWhat
is the
probability
a company The
at function:
EX 4: Suppose a (b)
certain
of small
data processing
firm is sochosen
specialized
that some have
at
random
spends
more
than
50%
of
its
budget
on
random
expends
less
than
10%
of
its
budget
on
enity density
function
thatofcharacterizes
the proportion
3.36
difficulty making
a profit
in
their
first
year
operation.
The
probability
density
function
that On xa labo
vironmental
and
pollution
controls?
(a) Find−6the
6 proba
environmental
and
pollution
controls?
Y the
that
makeYathat
profit
isa profit
givenis by
characterizes
proportion
make
given
by
f (x) = e the
, for
x
working,
den
(c) What is the probability that a company selected
time period,
m
x!
!
X,intersection.
is
4
3than 50% of its budget on
at random ky
spends
more
(1
−
y)
,
0
≤
y
≤
1,
3.33 fenvironmental
Suppose
a
certain
type
of
small
data
processing
(a)
Find
the
probability tha
and pollution controls?
(y) =
(b)period,
Find more
the proba
firm is so specialized
that someelsewhere.
have difficulty making time
0,
than 8
intersection.
a
profit
in
their
first
year
of
operation.
The
probabilf
(x)
=
a certain
of asmall
processing
a) What is the3.33
value Suppose
of k that renders
thetype
above
valid data
density
function?
Sol: 280
(a)ityWhat
the
valuethat
of ofsome
kcharacterizes
that
the
above
(b) aFind
the probability
tha
firm
is is
sothat
specialized
that
haverenders
difficulty
making
density
function
proportion
3.36
On a labora
b) Find the
probability
at most 50%
the firms
make
athe
profit
in the first year.
a
profit
in
their
first
year
of
operation.
The
probabilvalid
density
function?
Y that make a profit is given by
Sol: 0.3633 the densit
ity density function that characterizes the proportion 3.36 working,
On
(a)aislaboratory
Calculateassign
P(
!
c) Find(b)
the Find
probability
that
at
least
80%
of
the
firms
make
a
profit
in
the
first
year.
the
probability
that
at
most
50%
of
the
firms
X,
4 is given3 by
Y that make a profit
working,
the
density
function
ky (1 − y) , 0 ≤ y ≤ 1,
Sol: 0.0563
make fa(y)
profit
year.
= !in 4the first
X, is (b) What is the!
ky (1 − y)3 , 0 ≤
y ≤ 1,
0,
elsewhere.
f (y)
=
that
X
(c) Find the
probability
that at
least 80% of the firms (c) Given
!f (x)
0,
elsewhere.
=x),
2(1
−
X=will be les
make
a profit
the of
first
year.renders the above a
f (x)
(a)
What
is the in
value
k that
0,
(a) What is the value of k that renders the above a
valid
density function?
valid density function?
(a) Calculate
P (X
(b) (b)
Find
the
thatatatmost
most
50%
of firms
the firms
(a) Calculate
P (X ≤ 1/3).
Find
theprobability
probability that
50%
of the
(b) isWhat
is the prt
make
a aprofit
firstyear.
year.
(b) What
the probability
make
profit in
in the
the first
3.4
Joint Probability Distributions
f (x) =
, for
thetobaccos.
weight of
her
Thethe
propor- (b) Find
the accounts
probability
first
is busy more
tobacco
forthat
overthe
half
theline
blend.
x!
stic in a blend are random
than
75%
of the time.
Findamount
the marginal
density function
for the propor3.47(b)The
of kerosene,
in thousands
of liters,(a) Determine the probabili
function (X = Turkish and
tion of the domestic tobacco.
eight of the toffees in a tank
4, 5, and 6.
at the
the number
beginningphone
of any
day
is a by
random
3.65 Let
received
a
(c) Find the
probabilityofthat
the calls
proportion
of Turkgram if it is known amount
Y fromduring
whichaa5-minute
randominterval
amountbeXa israndom
sold dur-(b) Graph the probability m
switchboard
x, yweight.
≤ 1, x + y ≤ 1,
ish tobacco is less than 1/8 if it is known that the
he
variable
X with
probability
function
ing that
day.
Suppose
that
the
tank
is not resupplied ues of x.
where.
blend
contains
domestic
tobacco.
EX 5: Let X denote the
diameter
of an3/4
armored
electric
cable and Y denote the (c)
diameter
of the
Determine the cumulativ
during
the the
daycable.
so−2that
xand
≤ Yy,areand
assume
thatrange
thebetween
x
ceramic
mold
that
makes
Both
X
scaled
so
that
they
and of X.
e 2
at
a given
the Turkish
these 0values
thsin of
life, box
in1.years,
f
(x)
=
,
for
x
=
0,
1,
2,
.
.
.
.
joint
density
function
of
these
variables
is
company offers its policyholders a
Suppose that X3.62
and YAn
haveinsurance
the joint
ver half the blend.
x! density
system. If the joint
number
of
different
premium
payment options. For a
!
ity function for the propor2,
0
<
x
≤letyXX<equals
1,the 0,number
(a)
Determine
the
probability
that
1, 2, 3,of 3.66 Consider the random
sbacco.
randomly
selected
policyholder,
be
/ /
joint density function
4, f
5,(x,
andy)6.= successive
months
between
payments. The cumulative
0, elsewhere.
hat the proportion of Turk- distribution function
!
of Xmass
is function for these valy1/8
> if0,it is known that the (b) Graph the probability
x + y,
⎧
f (x, y) =
uesand
of Yx.are
a)
Determine
if
X
independent.
⎪
0,
if
x
<
1,
(a)
Determine
if
X
and
Y
are
independent.
mestic
⎪
0,
ere, tobacco. Chapter 3 Random Variables
⎪
and Probability
Distributions Sol: 1/3
⎪
b) Find P(¼ (c)
< XDetermine
< ½ | Y = ¾the
) (!cumulative
⎪
0.4, if 1distribution
≤Probability)
x < 3, function for
⎨Conditional
these values
of X.
(a) Find the marginal distri
F (x) =
0.6, if 3 ≤ x < 5,
ny offers its policyholders a
⎪
⎪
um
payment options. For a 3.53 Given the joint
⎪0.8,density
(b) Find P (X > 0.5, Y > 0.
if 5 ≤ function
x < 7,
= 3/4).
Consider
the⎪
⎪
EX number
6: Givenofthe 3.66
joint density
function
⎩random variables X and Y with
lder, let X be the
1.0, if x ≥ 7.
joint density function
! 6−x−y
payments. The cumulative
/ 4,/
3.67 An industrial proces
, 0 <mass
x <function
2, 2 < yof <
!
9,
8
(a)f What
is
the
probability
X?
is find
(x, y) =
can be classified as either
x + y, 0 ≤ x, y ≤ 1,
f (x, y)P0,
=
(b) Compute
(4 < X ≤ elsewhere,
7).
The probability that an ite
f x < 1,
0,
elsewhere.
find
P(
1
<
Y
<
3
|
X
=
1)
(!
Conditional
Probability).
5/8 is conducted
experiment
in
f 1 ≤ x < 3,
108
Chapter
3Sol:Random
Variables
find
P
(1
<
Y
<
3
|
X
=
1).
randomly
from
the
process.
3.63
Two
components of X
a missile
(a) Find
theelectronic
marginal distributions
and Y . system
f 3 ≤ x < 5,
the number
work
in
harmony
for
the
success
of
the
total
system.
EX
7:
Two
electronic
components
of
a
missile
system
work
in
harmony
for thebesuccess
of the of defectives
(b) Find P (X > 0.5, Y > 0.5).
fr 5of≤times
x < 7,a certain nuthe
probability
mass
Let
XDetermine
Y its
denote
lifethe
in
of
the two
comwhether
two
random
variables
of
total system.(a)
Let3.54
X and
Yand
denote
the
lifethe
in hours
ofhours
the two
components.
The is
joint
density
of time
3.74
The
Z func
in m
Determine
probability
density
function.
f x ≥ 7.
ponents.
The
joint
density
of
X
and
Y
is
malfunction:
1,
2,
or
3
X and Y is
Exercise
3.49
are
dependent
or
independent.
trical
supply system h
An industrial
manufactures
(b) 3.67
Determine
the probability
that the lifeitems
spanthat
of such3.68
! process
Consider the followin
the number
−y(1+x)
y denote
mass function
of X? of
tion
,
x,
y
≥
0,
ye
can
be
classified
as
either
defective
or
not
defective.
a component
exceed 70 hours.
function of the random vari
f (x, y) =will
n).emergency call. Their 3.55
Determine
whether
the
two
random
variables
of
!1 −
The probability
that
iselsewhere.
defective
is 0.1.
An
0, an item
! 3x−y
e
given as
experiment
is are
conducted
in which
5 items are drawn
Exercise
3.50
dependent
or independent.
1<
f (z)9= , 10
f
(x,
y)
=
a)
Give
the
marginal
density
functions
for
both
random
variables.
Sol:
?
(a)
Give
the
marginal
density
functions
for
both
ran3.70
Pairs
of
pants
are
being
produced
by
a
particux
0,
ponents of a missile system randomly from the process. Let the random variable X
0,
elsew
dom
variables.
b) system.
Whatlar
is the
probability
that
the
lives
of both
components
will
exceed 2ofhours?
outlet
facility.
The
pants
are
checked
by
a
group
be
the
number
of
defectives
in
this
sample
of
5.
What
2
success
of the3 total
3.56 The joint density function of the random vari6
is the
mass function
X?
(b)
What
isThe
the
probability
thatofthe
livesof
of pants
both com10ables
workers.
inspect
pairs
taken(a) Sol:
(a)
What
is the densi
prob
fe
in hours 0.10
of the two com0.05
Find1/(3e
the )marginal
Xprobability
and
Y workers
is
ponents
will
exceed
2
hours?
of X and Y0.35
is
randomly from the production line. Each inspector is(b) Arewithin
0.10
X and a
Y 20-minut
independe
! the
3.68
Consider
following
joint
probability
density
EX
8:
Consider
the
following
joint
probability
density
function
of
the
random
variables
X
and
x)
assigned
a
number
from
1
through
10.
A
buyer
selects
0.20
0.10
6x,
0
<
x
<
1,
0
<
y
<
1
−
x,
is 2).
the prob
, x, y ≥ 0,
FindWhat
P (X >
function
random
variables
X
and
Y :service lines. (c) (b)
Y:
f (x,
y)of
=thefor
3.64
service
facility
operates
with
two
a pair
of A
pants
purchase.
Let
the
random
variable
within
10
minutes
elsewhere.
elsewhere.
ibution of X.
!0,
Onthe
a randomly
selected
day,
let
X 1be
the
proportion
of
3x−y
X be
inspector
number.
,
1
<
x
<
3,
<
y
<
2,
9 line is in use whereas Y is the pro3.69 The life span in hou
that
y)the
= first
ibution
of Y .for both ran- timef (x,
sity functions
0,
elsewhere.
(a)(a)
Give
a
reasonable
probability
mass
for X.nent3.75
is a random
variable
w
Showofthat
and
Y second
are not
independent.
portion
timeX
that
the
line
is infunction
use. Suppose
A chemical
syst
function
that
the
joint
probability
density
function
for
(X,
Y
)
is
(b)
Plot the
cumulative
for X.
reaction
has two impo
a) Find
the(b)
marginal
functions
of=X 0.5).
and
Y. function
Sol:
?
y that the lives of both
comP
(Xmarginal
>
0.3 |density
Ydistribution
(a)Find
Finddensity
the
functions
of X and Y .
!
!
x
2
2
− 50
in
a
blend.
The
3
ours?
b) Are X and(b)
Y independent?
Sol: ?
1 − ejoint
(x
+
y
),
0
≤
x,
y
≤
1,
,
Are
X
and
Y
independent?
2
have the following
joint
F (x)X
=1 and X2 o
f (x, y) =
portions
c) Find P(
X
>
2).
Sol:
2/3
3.71
The
shelf
life
of
a
product
is
a
random
variable
0,
0,
3.57
LetP X,
Y ,2).
and Z have elsewhere.
the joint probability den(c) Find
(X >
by
erates with two service lines.
that
is
related
to
consumer
acceptance.
It
turns
out
sity of
function
x let X be the EX
9: The shelf
ashelf
product
variable
that is related
to
y,
proportion
of
!
thatlife
the
lifeisspan
Ya random
inindays
of ofa certain
typecompoofconsumer
bakery acceptance. It
The
life
hours
an
electrical
2
n use4 whereas turns
Y is out
the that
pro-the3.69
!
shelf
life
Y
in
days
of
a
certain
type
of
bakery
product
has
a
density
function
2 function
product
has
a density
f (x1 , x2 ) =
nent
is
a
random
variable
with
cumulative
distribution
kxy
z,
0
<
x,
y
<
1,
0
<
z
<
2,
cond
line
is
in
use.
Suppose
0.15
0
f (x, y, z) = !
ensity
0,1 −y/2 elsewhere.
0.30function for (X, Y ) is function
e
,x 0 ≤ y < ∞,
(a) Give the marginal
f (y) = !21 − e− 50
), 0.15
0 ≤ x, y ≤ 1,
, x > 0,
0,
elsewhere.
F
(x)
=
(a) Find k.
(b) Give the marginal
0,
eleswhere.
ion elsewhere.
of X.
1
(b) Find
P (Xof<the
, Y > 1 , 1 < Z < 2).
(c) What is the prob
What
fraction
4 loaves2 of this product stocked toWhat fractionday
of the
loaves
of
this
product
stocked
today
would
you
expect
to
be
sellable
3 daysproduce the
tions
would you expect to be sellable 3 days from now?
from now?
without replacement 3.58 Determine whether the two random variables of (d) Give the condition
Passenger
congestion
is aorservice
problem in airExercise
3.43 are
dependent
independent.
queens, and kings) of 3.72
cards. Let X be the ports. Trains are installed within the airport to reduce 3.76 Consider the si
congestion.
With
the usethe
of the
the variables
time X inof
Determine
whether
twotrain,
random
Y the number of jacks. the3.59
But suppose the joint
minutes
that
it takes
to travel or
from
the main terminal
Exercise
3.44
are dependent
independent.
tions is given by
to a particular concourse has density function
!
ution of X and Y ;
3.60 The joint probability
density function of the ran6x
!1
f (x1 , x2 ) =
s the region given by dom variables X, Y10, ,and0 Z
≤ is
x ≤ 10,
0,
f (x) =
0,
elsewhere.
"
4xyz 2
(a) Give the marginal
, 0 < x, y < 1, 0 < z < 3,
9
ion of Y .
n?
/
/
⎩
0,
elsewhere.
117
que jewelry dealer is interthe average
number
of hours
per year
thatthe
families
random
variable
Y = 3X
− 2, where
X has
density
on pagefor
120which
to find
vari- Find
necklace
thethe
probrun
their
vacuum
cleaners.
able
of Exercise
4.7 onthat
page function
127
, andX0.14,
respectively,
!
or a profit of $250, sell it for 4.14 Find the proportion
1 −x/4
who can be
e X, of xindividuals
>0
4
n, or sell it for aEX
loss
of
$150.
f
(x)
=
expected
to
respond
to
a
certain
mail-order
solicitation
10:
Find
the
proportion
X
of
individuals
who
can
be
expected
to respond to a certain mail
losses, what
premium
ion
of X, the
number
of 0.1. Ignoring all other0, partial elsewhere.
m
with
the following
t? variable
if
X
has
the
density
function
order
solicitation
if
X
has
the
density
function
a synthetic fabric in con- should the insurance company charge each year to re% variance
an average
profit
of $500? of the random variable Y .
given
in Exercise
3.13for alizeFind
the mean
and
esis to
insure
airplane
−2
3 120
5his
< x <X1,has the density
3X
−, 2,0where
on page
to find the vari- random variable Y =2(x+2)
5
ompany
estimates
that
a
tof
(x)
=
.3 X 0.2
0.5 4.7 on page function
ble
of Exercise
0,
elsewhere.
4.44
Find
the
covariance
of
the random
variables
experts examine
stacks
of tiresX
obability
2tion
3 0.002,4 a 50% loss 4.10 Two tire-quality
of X.
!
and Y of Exercise
3.39
page0 105.
−x/4 onto
a 25%0.05
loss with
quality14 erating
tire on a 3-pointSol: 8/15
, x >each
16
0.01probability and assign fa(x)
=
scale.
Let
X
denote
the
rating
given
by expert A and
m
variable
with
the
following
ble X, representing
0,
elsewhere.
mperfections
per 10 the
me-num- 4.45 Find the covariance
of
theThe
random
variables
Y
denote
the
rating
given
by
B.
following
tableX
nes of softwareEX
code,
has proportion
the and of
11: The
people
who respond
to
a certain
mail order solicitation is a random
Y
of
Exercise
3.49
on
page
106.
gives
the
X and
Y . variable Y .
Find
thejoint
meandistribution
and variancefor
of the
random
2tribution:
3
5 variable X having
the density function given in EX 10. Find the variance of X.
3of
0.2
0.5discrete
of 4the
3ionRandom
5 Variables
6 ran119Sol:
4.46Find
Find
covariance
ofy the
random
variables
X 37/450
4.44
thethe
covariance
of the
random
variables
X
f
(x,
y)
1
2105.105. 3
25 of 0.4
on
X. 0.3 0.04
and
Exercise
3.44
page
and
Y Yof of
Exercise
3.39
on on
page
1 respond
0.10 to0.05
0.02
The proportion
of people who
a certain
mail order solicitation is a random
page
121, find EX
the 12:
variance
of
le X, representing the numx
2
0.10
0.35
0.05
What
is
the
population
mean
of
the
times
repair?
4.47
For
the
random
variables
X
andvariables
Ytowhose
4.45
Find function
the covariance
ofEX
the10.
random
Xjoint
, x = 0, 1, 2,variable
3.
X
having
the
density
given
in
es of software code, has the and
2
3 3.49
0.03
Y of
106. 0.20
density
function
is+given
in 0.10
Exercise
3.40 on page 105,
Find theinExpected value
of Exercise
g(X)
= 3X
4.on page
Sol: 5.1
,ibution:
or mean weight,
find
the
covariance.
probabilities are 0.4, 0.3, 4.31
0.2,
Consider
Exercise
3.32
on
page
94.
and
µ
.
Find
µ
X
Y
5 3 power
6
at 0, 41, 2, or
failures 4.46 Find the covariance of the random variables X
(a) What
isGiven
the mean
allo5
0.4
0.3
0.04
nswer
in
(b)?
Explain
4.48
a random
X,the
withbudget
standard
deY of
Exercise
3.44proportion
on variable
page 105.of
ivision in any given year. Find and
4.11
The
density
function
of
coded
measurements
of threads of a fitting
EX
13:
The density
function
of
coded
measurements
of
the
pitch
diameter
of
ndom
variable
T
reprecated
to
environmental
and
pollution
control?
viation
σ
,
and
a
random
variable
Y
=
a
+
bX,
show
X
age
find variable
the variance
of
the121,
random
X repretheWhat
pitch
threads
ofcoefficient
aafitting
is 3.25this
oins failures
in Exercise
on sub4.47
the
variables
X
and
Y ρis
whose
joint
that For
ifisdiameter
b the
< 0,random
theofcorrelation
−1, and
(b)
probability
that
company
selected
XY =
ower
striking
%
density
function
is
given
in
Exercise
3.40
on
page
105,
if
b
>
0,
ρ
=
1.
at randomXYwill have
allocated to environmental
dealt
with an impor4
,
0
<
x
<
1,
find the
covariance.
robabilities are 0.4, 0.3, 0.2, and
π(1+x2
pollution
a) proportion that exceeds the
f (x) = control
aracterized
t in
0,a units
1,
2, or
powertimes
failures
at
head
is 3by
three
4.49 Consider
the
situation
in Exercise 4.32 on page
of
$5000,
on
a new population
0,
mean
given
in elsewhere.
(a)?
Given
a random
variable
X,
with of
standard
devision
inexpected
any
givennumber
year.
Find 4.48
nd
the
119.
The
distribution
of
the
number
imperfections
variable
X
having
the
density
1,
viation
σ
,
and
a
random
variable
Y
=
a
+
bX,
show
X
the
random
variable
X
repreexpected
value10
of meters
X
Sol: ln4/π
ofvalue
synthetic
failure is given by
dsetwice.
4.12 on pageFind
117.theFind
the
Findper
thebexpected
of X.
< 0, the 3.13
correlation
coefficient
ρXY
= −1, and of
wer failures striking this subwhere.
4.32thatInif Exercise
on
page
92,
the
distribution
x
0
1
2
3
4
b > 0, ρof
XY = 1.
the if
number
imperfections
per
10
meters
of
synthetic
of the
number
of
imperfections
per
10
meters
of synthetic fabric is
woman is paidEX
$314:
if the
shedistribution
f (x) 0.41
0.16of $5000,
0.05 0.01
4.12 If a dealer’s
profit,0.37
in units
on a new
respond
toora cerfabric
is
given
by
given
by
5npeople
ifunits
she who
draws
a
king
4.49
thelooked
situation
in Exercise
4.32 of
onvariable
page
of $5000,
on aXnew
automobile
be
upon
asdeviation
a random
FindConsider
thecan
variance
and standard
the
numon
a random
variable
ofis52
playing
If hav- 119.
x
0
1
2
3
4
The
distribution
of
the
number
of
imperfections
ariable
X havingcards.
the density
X
having
the
density
function
ber
of
imperfections.
given
in
Exercise
4.14 on page per 10 meters of synthetic failure is given by
oses.
How
much
4.12 on
page
117.should
Find the
f(x) 0.41& 0.37 0.16 0.05 0.01
ffair?
X.
xprobability
0 2(1
1x),assignment,
94, the distribution
of
−function.
02< x <3 1,
4.50
For
a laboratory
if the 4equipmentSol:
is 0.88
(a)expected
Plot
the
a) Find the
number
of
imperfections.
f
(x)
=
f
(x)
0.41
0.37
0.16
0.05
0.01 outcome
washing
machine
was
function
of
the
observed
of
hours,
in
units
of
100
hours,
2 working, the density
0,
elsewhere,
people
who respond
atocerb) toFind
E(X
).
(b)
Find
expected number of imperfections,Sol: 1.62
ash
is cleaner
paid
according
X isthethe
over a period
of Find
variance
and standard deviation of the num- Sol: 0.8456
nuum
is
a
random
variable
X
havc) Find
Var(X)
E(X)
= µ.
through.
Suppose
the
ariable
X
having
the
density
! per automobile.
ber
of
imperfections.
profit
iven in
Exercise
4.14
on
pagefind the average
2
1/4,
1/4,
1/6,
and
1/6,
2(1 − x), 0 < x < 1,
0,
se
the Find E(X f).
X.4.13 on page 117. Find (c)
(x) =
nt
receives
$7,
$9,
$11,
0, assignment,
otherwise.
4.50The
Fordensity
a laboratory
if the equipment
is
where.
4.13
function
of the
continuous
random
EX
15:
For
a
laboratory
assignment,
if
the
equipment
is working,
the density function of the
P.M.
and
5:00
P.M.
on
working,
the
density
function
of
the
observed
outcome
f hours, in units of 100 hours,variable
X, the total number of hours, in units of 100
observed
outcome
endant’s
earnrcise
4.14expected
on
page
117,
find
XXFind
isis the variance and standard deviation of X.
uum
cleaner
over
a period
ofhours,
that
a family runs a vacuum cleaner over a pe2
X) = 3X
+ 4. the density
riable
X having
!
riod 4.51
of oneFor
year,
is
given
in 0Exercise
3.7 Yoninpage
92
2(1
− x),variables
<x<
the
random
X 1,and
Exercise
4.13 on page 117. Find the
(x) =
as 3.39 onfpage
105,
the correlation coefficient
0, determine
otherwise.
lar
stock, ofa the
person
can varideviation
random
⎧
between
X
and
Y
.
000
with probability
n Exercise
4.17Find
on page
118. and standard deviation
x,Xstandard
0<
x < 1, importance
⎨and
the 0.3
variance
ofisX.of
ciseor4.14
on page
117,
find
the variance
deviation
of X.
n,
expected
value,
of a Find
random
variable
special
inSol: 1/18, 0.2357
robability
0.7.
What
is
2
X)
=
3X
+
4.
f
(x)
=
4.52
Random
variables
X
and
Y
follow
a
joint
distri2
−
x,
1
≤
x
<
2,
because
describes
theFor
probability
is centered. By
of
Exerciseit4.21
on page where
118, 4.51
⎩ distribution
the
random
variables
X and Y in Exercise
bution
EX
16:
Random
variables
X
and
Y
follow
a
joint
distribution
2
0,
elsewhere.
wever,
mean
not give an adequate description
of the shape
of the
) = X the
, where
X isdoes
a random
the correlation
coefficient
!
deviation of the random vari- 3.39 on page 105, determine
ity
function
given
in
Exercise
on.
We
also
need
to
characterize
the
variability
in
the
distribution.
In
between X and Y .
jewelry dealer
inter2, 0 < x ≤ y < 1,
Exercise
4.17 on is
page
118. Find
the average
number
of hours per year that families
f
(x,
y)
=
which
thehistograms
prob1,lace
weforhave
the
of
twovacuum
discrete
probability
distributions that
0, otherwise.
run4.52
theirRandom
cleaners.
variables
X and Y follow a joint distrid
0.14,
respectively,
that
same
mean,
= page
2, but
considerably
in variability,
f Exercise
4.21µ
on
118,
Determine
thediffer
correlation
coefficient between
X and Y or the dispersion
bution
e,
in2minutes,
for
an
airplane
profit
of
$250,
sell
it
for
Determine
the
correlation
between
X be
and
=
X
,
where
X
is
a
random
bservations
about
the mean.
X ofcoefficient
individuals
who can
!
akeoff
is4.14
a Y .Find the proportion
sell
it at
fora acertain
loss of
$150.
ty
function
given
inairport
Exercise
2, 0 < x ≤ y < 1,
expected tofrespond
(x, y) = to a certain mail-order solicitation
0, otherwise.
if X has the density function
Covariance of Random Variables
, insure
in minutes,
for an airplane
his airplane
for
keoff
at
a
certain
airport
any estimates that
a to-is a
bility 0.002, a 50% loss
5% loss with probability
%
Determine the correlation
2(x+2) coefficient between X and
, 0 < x < 1,
5
Y.
f (x) =
0,
elsewhere.
2 − x, 1
≤ x < 2, −
valuate
E(2XY
⎩
0,
elsewhere.
tribution
shown in
X Y)
Table
green die is tossed and Y the number that occurs when
a certain
2-minute
period
day. ofThe
distria red die
is tossed.
Find in
thethe
variance
the joint
random
variable
bution is
%
&% &
o evaluate the mean of the random (a) 2X − Y ;
2
1
9
+ 39X, where Y is equal to the (b) X +f3Y
−y)
5. =
(x,
,
(x+y)
pendent
randomannually.
variables
t hours expended
16
4
/ /
σY2 = 3, find the variance
4.67
If
the
joint
density
function
of
X
and
Y
is given
EX 17:such
If the
joint density function of X and Y is given by
variable
X is−defined
that
for xby= 0, 1, 2, . . . and y = 0, 1, 2, . . . .
−2X
+ 4Y
3.
$ 2 E(Y ), Var(X), and Var(Y ).
(a) GiveExercises
E(X),
= 10 and E[(X − 2)2 ] = 6,
(x + 2y), 0 < x < 1, 1 < y < 2,
if X and Y are not inde- Review
f (x, y) = Z 7 = X + Y , the sum of the two. Find
(b) Consider
0,
elsewhere,
E(Z) and Var(Z).
(b) Itfind
is the
of interest
know
something
pro- Then do it not by usi
expected3to
value
of g(X,
Y ) = YX3 about
+ X 2 Y the
.
t are
X and
Y are independent
random
2
independent
random
find
the
expected
value
of
g(X,Y)=
(X/Y
)+X
Y.
46/63
portion
of
Z
=
X
+
Y
,
the
sum
of
the
two
proporthe first-order
Taylor se
he joint probability distribution 4.70 Consider Review Exercise 3.64 on page 107. Sol:
nsities and
tions.
Find
E(X
+
Y
).
Also
find
E(XY
).
4.68 The power P in watts which is dissipated in an
Comment!
x
There
are lines.
two
service
lines.
The
random
variables
X
EX
twoFind
service
The
random
variables
X
and
Y
are
the
proportions
of time
electric
circuit
with
resistance
R
is
known
to
be
given
x, y)
2
4 18: There are (c)
Var(X),
Var(Y ), and
Cov(X,
Y ).line 1 and line
2
and
Y
are
the
proportions
of
time
that
> 2,
by
P
=
I
R,
where
I
is
current
in
amperes
and
R
is
a
that line 1 and line 2 are in use, respectively. The joint probability density function4.75
for (X,An
Y) electrical firm
1
0.10 0.15
(d)
Find
Var(X
+atY50).ohms. The
/ / variconstant
fixed
However,
is a random
2 are
in use,
respectively.
jointI probability
density bulb, which, according
is given by
3
0.20 0.30
sewhere,
able with
µI =Y 15
and σI2 = 0.03 amperes2 .
function
for (X,
) isamperes
given by
5
0.10 0.15
package, has a mean lif
Give numerical approximations to the mean and vari4.71ance
The
length
of
time
Y
,
in
minutes,
required
to
deviation of 50 hours.
$
of the power
2
3 P2.
138
Chapter 4 M
+ yto
),tear
0 ≤gas
x, has
y ≤the
1, density the bulbs fail
generate
a human2 (x
reflex
to last eve
f
(x,
y)
=
< y < 1,
function
4.69 Consider0,
Review Exerciseelsewhere.
3.77 on page 108. The
distribution is symmetr
2and Y represent the number of verandom
1 X−y/4
sewhere.
Find
E(X)variables
and !E(X
) and
using
these values, 4.65 Let X represent the n
e
,separate
0then,
≤ ystreet
<
∞,corners
4 at
hicles fthat
2 twoor
2
2 (a)
(y) arrive
=whether
red4.76
die is Seventy
tossed and
Y jobs
the
Determine
not
X
and
Y areduring
indepennew
evaluate
E[(2X
+
1)
].
m 4.7 to evaluate E(2XY
−
X
Y
)
a) Determine whether
or not2-minute
X and0,Yperiod
are independent.
a certain
inelsewhere.
the day. The joint distri- a green die is tossed. Find
Zability
= XYdistribution
.
bile
manufacturing
plan
shown in Table dent.
bution
b) It is of interest to
knowissomething about the proportion of Z = X+Y, the sum of the
4.58
The
total
time, %
measured
of 100 hours, (a)for
E(X
Y );positions. To
the+70
&
%in units
(a)
What
isE(X+Y).
the mean
time
to&reflex?
two proportions.
Find
Sol: 5/4
1
9
that a teenager
runs
her
hair
dryer
over
a
period
of
one
the
applicants,
the com
2f (x, y) =
,
c) Alsovariables
find E(XY
). is aE(Y
Sol: (b)
3/8 E(X − Y );
(x+y)
(b)year
Find
) and Var(Y
). variable
are independent random
continuous
random
X
that
has
the
16
4
mechanical
skill,
manua
= 5 and σY2 = 3, find
the variance
d) Find
Var(X),density
Var(Y ),function
and Cov(X, Y).
Sol: 73/960, 73/960, -1/64 (c) E(XY ).
ability. The mean grad
for x = 0, 1, 2, . . . and y = 0, 1,Sol:
2, . .29/240
..
able Z = −2X + 4Ye) −Find
3. Var(X + Y).
⎧
4.72 A manufacturing
company has
developed a ma- 60, and the scores have
(a) Give E(X), E(Y
Var(Y ).
4.66 Let X represent the n
x, ), Var(X),
0 < xand
1,
chine for cleaning ⎨
carpet
that
is <
fuel-efficient
becausegreen
rcise 4.62 if X and Y are not indea person
who and
scores
84
is tossed
Y the
(b)
Consider
Z
=
X
+
Y
,
the
sum
of
the
two.
Find
EX
19:
A
manufacturing
company
has
developed
a
machine
for
cleaning
carpet
that die
is fuelf
(x)
=
2
−
x,
1
≤
x
<
2,
= 1.
it delivers
carpet
cleaner
so
rapidly.
Of
interest
is
a
jobs?
[Hint:
Use
Cheby
⎩
a red Y,
diethe
is tossed. Find th
and
Var(Z).
efficient because itrandom
deliversE(Z)
carpet
cleaner
so rapidly.
Ofininterest
a random
variable
elsewhere.
variable
Y , 0,
the
amount
gallonsis per
minute
the distribution is symm
variable
amount in gallons
per minute It
delivered.
It is
known
the density
function
is given by
t X and Y are independent
random
delivered.
is known
that
the that
density
function
is107.
given
(a) 2X − Y ;
Consider
Review
Exercise
3.64 ofonthe
page
Use4.70
Theorem
4.6 to
evaluate
the mean
random
bability densities and
byvariable
2
A−
random
variab
There
are
two
service
lines.
The
random
variables
X
Y
=
60X
+
39X,
where
Y
is
equal
to
the
(b)4.77
X + 3Y
$8
25.
!
and Y ofare
the proportions
of time that
line 1 and line
variance σ = 4. Using
, x > 2,
number
kilowatt
hours
x3
1, expended
7The
≤ yjoint
≤ 8,annually.
2 are in use,
respectively.
probability density
0,
elsewhere,
f (y)
=
(a) If
P (|X
− 10|density
≥ 3); fun
4.67
the joint
function
for (X, Yvariable
) is
0,given
elsewhere.
4.59
If a random
Xbyis defined such that
by (b) P (|X − 10| < 3);
$3 2
2
2
2y ≤ 1,
(x
+
y
),
0
≤
x,
(a)
Sketch
the
density
function.
$
E[(X
(c) P (5 $
<2X
15); 0 <
f (x,−y)1)=] =2 10 and E[(X − 2) ] = 6,
(x <
+ 2y),
2y, 0 < y <Give
1, E(Y), E(Y2), and Var(Y).
Sol:
7.5,
169/3,
1/12
2
0,
elsewhere.
7
f(d)
(x, y)
= value of the con
(b) Give E(Y 2), E(Y ), and Var(Y ).
the
0,
els
0,
elsewhere.
find µ and σ .
whether random
or not Xvariables
and Y are
indepenEX 20: Suppose that X(a)
andDetermine
Y are independent
having
the joint probability
dent.
4.73
For
the situation
inY are
Exercise
4.72, random
computefind the expected value of g(
4.60
Suppose
that
X
and
independent
Y
P (|X −
E(e
) using
Theorem
4.1,probability
that is, bydistribution
using
variables
having
the joint
" 8
4.68 The power P in watts
x
Y
y
electric circuit with resistan
4
=y) e 2f (y) dy.
E(e f)(x,
2
value of Z = XY
.
distribution
by P = I R, where I is curr
1 7 0.10 0.15
constant
at 50 ohms.
H
4.78 fixed
Compute
P (µ −
y
3
0.20 0.30
Then compute E(eY ) 5not by
using
f
(y),
but
rather
by
able
with
µ
=
15
amperes
I
has the density function
0.10 0.15
using the second-order adjustment to the first-orderGive numerical approximati
Find
ance of the power P .
approximation
of E(eY ). Comment.
Find
(a) E(2X − 3Y );
!
6x(1
4.69
Consider
Review
Exer−
(b) E(XY
).
4.74
Consider
again the situation of Exercise 4.72. It
f (x) =
a) E(2X − 3Y )
Sol: -2.60
0, Y r
is required to find Var(eY ). Use Theorems 4.2 and 4.3random variables X and
b) E(XY )
Sol:
9.60
hicles
that
arrive
at
two
sepa
4.61
UseZTheorem
4.7 to use
evaluate
E(2XY 2 − of
X 2ExerY ) a certain 2-minute period in
the conditions
and
define
= eY . Thus,
for4.73
the to
joint
probability distribution shown in Table bution is
cise
find
and compare with the r
3.1 on page 96.
%
theorem.
Var(Z) = E(Z 2 ) − [E(Z)]2 .
1
f (x, y) =
(x+y)
4.62 If X and Y are independent random variables
4
2
with variances σX
= 5 and σY2 = 3, find the variance
Review
Exercises
for x = 0, 1, 2, . . . and y = 0
of the random variable Z = −2X + 4Y − 3.
(a) Give E(X), E(Y ), Var(X
4.63 Repeat Exercise 4.62 if X and Y are not inde(b) Consider Z = X + Y , t
pendent
andChebyshev’s
σXY = 1.
4.79
Prove
theorem.
4.81 Referring to the
E(Z) and Var(Z).
probability density fun
Review Exercises
4.79 Prove Chebyshev’s theorem.
4.81 Referring to the r
probability
EX 21: Find the covariance of random variables X and Y having the joint probability
density density funct
4.80 Find the covariance of random variables X and on page 105, find the ave
function
in the tank at the end of
Y having the joint probability density function
!
x + y, 0 < x < 1, 0 < y < 1,
4.82 Assume the length
f (x, y) =
0,
elsewhere.
lar type of telephone conv
Sol: -1/144
EX 22: As a student drives to school, he encounters a traffic signal. This traffic signal stays
green for 35 seconds, yellow for 5 seconds, and red for 60 seconds. Assume that the student
goes to school each weekday between 8:00 and 8:30 a.m. Let X1 be the number of times he
encounters a green light, X2 be the number of times he encounters a yellow light, and X3 be the
number of times he encounters a red light. Find the joint distribution of X1, X2, and X3.
Sol: ?
EX 23: Suppose that for a very large shipment of integrated-circuit chips, the probability of
failure for any one chip is 0.10. Assuming that the assumptions underlying the binomial
distributions are met, find the probability that at most 3 chips fail in a random sample of 20.
Sol: 0.8670
EX 24: If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9,
find the probabilities that among 20 such lights
a) exactly 18 will have a useful life of at least 800 hours;
b) at least 15 will have a useful life of at least 800 hours;
c) at least 2 will not have a useful life of at least 800 hours.
Sol: 0.2852
Sol: 0.9887
Sol: 0.6083
EX 25: A manufacturer knows that on average 20% of the electric toasters produced require
repairs within 1 year after they are sold. When 20 toasters are randomly selected, find
appropriate numbers x and y such that
a) the probability that at least x of them will require repairs is less than 0.5;
Sol: 4
b) the probability that at least y of them will not require repairs is greater than 0.8.
Sol: 14