joint probability distributions

Chapter4TitleandOutline1

JointProbabilityDistributions

51TwoorMoreRandomVariables51.1JointProbabilityDistributions51.2MarginalProbability

Distributions51.3ConditionalProbability

Distributions51.4Independence51.5MoreThanTwoRandom

Variables

52CovarianceandCorrelation53CommonJointDistributions53.1MultinomialProbabilityDistribution53.2BivariateNormalDistribution54LinearFunctionsofRandomVariables55GeneralFunctionsofRandomVariables

CHAPTEROUTLINE

JohnWiley&Sons,Inc. AppliedStatisticsandProbabilityforEngineers,byMontgomeryandRunger.

Outline

JointProbabilityMass/DistributionFunction MarginalProbabilityDistribution Mean Variance ConditionalProbability Independence Covariance Correlation

Sec2 3


ConceptofJointProbabilities Somerandomvariablesarenotindependentofeachother,i.e.,theytendtoberelated. Urbanatmosphericozoneandairborneparticulatemattertendtovarytogether.

Urbanvehiclespeedsandfuelconsumptionratestendtovaryinversely.

Thelength(X)ofainjectionmoldedpartmightnotbeindependentofthewidth(Y).Individualpartswillvaryduetorandomvariationinmaterialsandpressure.

Ajointprobabilitydistributionwilldescribethebehaviorofseveralrandomvariables,say,X andY.Thegraphofthedistributionis3dimensional:x,y,andf(x,y).

Chapter5Introduction 4


Example51:SignalBarsYouuseyourcellphonetocheckyourairlinereservation.Theairlinesystem

requiresthatyouspeakthenameofyourdeparturecitytothevoicerecognitionsystem. LetYdenotethenumberoftimesthatyouhavetostateyourdeparture

city. LetXdenotethenumberofbarsofsignalstrengthonyoucellphone.

Sec51.1JointProbabilityDistributions 5

Figure51JointprobabilitydistributionofXandY.Thetablecellsaretheprobabilities.Observethatmorebarsrelatetolessrepeating.

1 2 31 0.01 0.02 0.252 0.02 0.03 0.203 0.02 0.10 0.054 0.15 0.10 0.05

x=numberofbarsofsignalstrength

y=numberoftimescity

nameisstated

Y1

Y2

Y3Y4

0

0.05

0.1

0.15

0.2

0.25

12

3x

Y1

Y2

Y3

Y4


JointProbabilityMassFunctionDefined


The of the discrete random variables and , denoted as , , satifies:

(1) , 0 All probabilities

joint probabilit

are non-negative

(2) , 1

y mass function

XY

XY

XYx y

X Yf x y

f x y

f x y

The sum of all probabilities is 1

(3) , , (5-1)XYf x y P X x Y y


0.15 0.10 0.05

0.02 0.10 0.05

0.02 0.03 0.20

0.01 0.02 0.25

0 1 2 3

1

2

3

4

Y = fY(y)

4

3

2

1

MarginalProbabilityDistributions(discrete)

Sec2 8

x

y

0.15 0.10 0.05

0.02 0.10 0.05

0.02 0.03 0.20

0.01 0.02 0.25

0 1 2 3

1

2

3

4

x

y

X = 1 2 3fX(x)

Xy

Yx

f x f xy

f y f xy


Y = fY(y)

4 0.30

3 0.17

2 0.25

1 0.28

MeanandVariance

Sec2 10

0.15 0.10 0.05

0.02 0.10 0.05

0.02 0.03 0.20

0.01 0.02 0.25

0 1 2 3

1

2

3

4

x

y

X = 1 2 3fX(x) 0.20 0.25 0.55

E(X)=E(X2)=V(X)=

E(Y)=E(Y2)=V(Y)=


JointProbabilityDensityFunctionDefined


(1) , 0 for all ,

(2) , 1

(3) , , (5-2)

XY

XY

XYR

f x y x y

f x y dxdy

P X Y R f x y dxdy

Figure52JointprobabilitydensityfunctionfortherandomvariablesX andY.Probabilitythat(X,Y)isintheregionR isdeterminedbythevolume offXY(x,y)overtheregionR.

ThejointprobabilitydensityfunctionforthecontinuousrandomvariablesX andY,denotesasfXY(x,y),satisfiesthefollowingproperties:


JointProbabilityMassFunctionGraph


Figure53JointprobabilitydensityfunctionforthecontinuousrandomvariablesX andY ofdifferentdimensionsofaninjectionmoldedpart.Notetheasymmetric,narrowridgeshapeofthePDFindicatingthatsmallvaluesintheX dimensionaremorelikelytooccurwhensmallvaluesintheYdimensionoccur.


Example52:ServerAccessTime1LettherandomvariableX denotethetime(msecs)untila

computerserverconnectstoyourmachine.LetY denotethetimeuntiltheserverauthorizesyouasavaliduser.X andYmeasurethewaitfromacommonstartingpoint (x


Example52:ServerAccessTime2 Thejointprobabilitydensityfunctionis:

Weverifythatitintegratesto1asfollows:


0.001 0.002 0.002 0.0010 0

0.0020.001 0.003

0 0

,

0.0030.002

10.003 10.003

x y y xXY

x x

xx x

f x y dxdy ke dy dx k e dy e dx

ek e dx e dx

0.001 0.002 6, for 0 and 6 10x yXYf x y ke x y k


Example52:ServerAccessTime2Nowcalculateaprobability:


Figure55RegionofintegrationfortheprobabilitythatX


MarginalProbabilityDistributions(continuous)

RatherthansummingadiscretejointPMF,weintegrateacontinuousjointPDF.

ThemarginalPDFsareusedtomakeprobabilitystatementsaboutonevariable.

IfthejointprobabilitydensityfunctionofrandomvariablesX andY isfXY(x,y),themarginalprobabilitydensityfunctionsofX andY are:

Sec51.2MarginalProbabilityDistributions 17

,

, (5-3)

X XYy

Y XYx

f x f x y dy

f y f x y dx


Example54:ServerAccessTime1FortherandomvariablestimesinExample52,findtheprobabilitythatYexceeds2000.

IntegratethejointPDFdirectlyusingthepicturetodeterminethelimits.


20000 2000 2000

2000 , ,

Dark region left dark region right dark region

XY XYx

P Y f x y dy dx f x y dy dx


Example54:ServerAccessTime2

Alternatively,findthemarginalPDF andthenintegratethattofindthedesiredprobability.


0.001 0.002

0

0.002 0.001

0

0.0010.002

0

0.0010.002

3 0.002 0.001

0.001

10.001

6 10 1 for 0

yx y

Y

yy x

yxy

yy

y y

f y ke

ke e dx

eke

eke

e e y

2000

3 0.002 0.001

2000

0.002 0.0033

2000 2000

4 63

2000

6 10 1

6 100.002 0.003

6 10 0.050.002 0.003

Y

y y

y y

P Y f y dy

e e dy

e e

e e

dx


Mean&VarianceofaMarginalDistribution

MeansE(X)andE(Y)arecalculatedfromthediscreteandcontinuousmarginaldistributions.


2 2 2 2

2 2 2 2

Discrete Continuous

X X XR R

Y Y YR R

X X X XR R

Y Y Y YR R

E X x f x x f x dx

E Y y f y y f y dy

V X x f x x f x dx

V Y y f y y f y dy


0.15 0.10 0.05

0.02 0.10 0.05

0.02 0.03 0.20

0.01 0.02 0.25

0 1 2 3

1

2

3

4

ConditionalProbabilityDistributions

Sec2 21

x

y Recall that P A B

P B AP A

FromExample51P(Y=1|X=3)=P(Y=2|X=3)=P(Y=3|X=3)=P(Y=4|X=3)=

Y = 1 2 3 4f (y|X=3)


ConditionalProbabilityDensityFunctionDefined

Sec51.3ConditionalProbabilityDistributions 23

Given continuous random variables and with joint probability density function , , the conditional probability densiy function of given =x is

, for 0

XY

XYXY x

X

X Yf x y

Y Xf x y

f y f xf x

(5-4)

which satifies the following properties:(1) 0

(2) 1

(3) for any set B in the range of Y

Y x

Y x

Y xB

f y

f y dy

P Y B X x f y dy


0.15 0.10 0.05

0.02 0.10 0.05

0.02 0.03 0.20

0.01 0.02 0.25

0 1 2 3

1

2

3

4

Mean&VarianceofConditionalRandomVariables

Sec2 27

x

y

Y = 1 2 3 4f (y|X=3) 0.455 0.364 0.091 0.091

FromExample51P(Y=1|X=3)=0.25/0.55=0.455P(Y=2|X=3)=0.20/0.55=0.364P(Y=3|X=3)=0.05/0.55=0.091P(Y=4|X=3)=0.05/0.55=0.091

E(Y|X=3)=

V(Y|X=3)=


JointRandomVariableIndependence

RandomvariableindependencemeansthatknowledgeofthevaluesofXdoesnotchangeanyoftheprobabilitiesassociatedwiththevaluesofY.

X andY varyindependently. DependenceimpliesthatthevaluesofX areinfluencedbythevaluesofY.

Doyouthinkthatapersonsheightandweightareindependent?

Sec51.4Independence 31


Exercise510:IndependentRandomVariablesInaplasticmoldingoperation,

eachpartisclassifiedastowhetheritconformstocolorandlengthspecifications.


1 if the part conforms to color specs0 otherwise

1 if the part conforms to length specs0 otherwise

X

Y

Figure510(a)showsmarginal&jointprobabilities,fXY(x,y)=fX(x)*fY(y)

Figure510(b)showtheconditionalprobabilities,fY|x(y)=fY(y)


PropertiesofIndependenceForrandomvariablesX andY,ifanyoneofthefollowingpropertiesistrue,theothersarealsotrue.ThenXandY areindependent.


(1) ,(2) for all x and y with 0

(3) for all x and y with 0

(4) P , for any sets and in the range of and , respectively. (5-7)

XY X Y

Y XY x

X YX y

f x y f x f y

f y f y f x

f y f x f y

X A Y B P X A P Y BA B X Y


RectangularRangefor(X,Y)

ArectangularrangeforX andY isanecessary,butnotsufficient,conditionfortheindependenceofthevariables.

IftherangeofXandYisnotrectangular,thentherangeofonevariableislimitedbythevalueoftheothervariable.

IftherangeofXandYis rectangular,thenoneofthepropertiesof(57)mustbedemonstratedtoproveindependence.



Covariance

Covarianceisameasureoftherelationshipbetweentworandomvariables.

First,weneedtodescribetheexpectedvalueofafunctionoftworandomvariables.Leth(X,Y)denotethefunctionofinterest.

Sec52Covariance&Correlation 49

, , for , discrete

, (5-13), , for , continuous

XY

XY

h x y f x y X YE h X Y

h x y f x y dxdy X Y


Example519:E(Functionof2RandomVariables)


Figure512DiscretejointdistributionofXandY.

Task:CalculateE[(XX)(YY)]=covariance

x y f(x,y) xX yY Prod1 1 0.1 1.4 1.0 0.141 2 0.2 1.4 0.0 0.003 1 0.2 0.6 1.0 0.123 2 0.2 0.6 0.0 0.003 3 0.3 0.6 1.0 0.181 0.3 0.203 0.7

1 0.32 0.43 0.3

X = 2.4

Y = 2.0Me

a

n

covariance=

J

o

i

n

t

M

a

r

g

i

n

a

l


***CovarianceDefined


The covariance between the random variables X and Y, denoted as cov , or is

(5-14)

The units of are units of times units of .

For example, if the units of

XY

XY X Y X Y

XY

X Y

E X Y E XY

X Y

X are feet and the units of Y are pounds,the units of the covariance are foot-pounds.

Unlike the range of variance, - .XY


Proof )

Note,however,that , ,

Similarly,itcanbeshownthat ,thus

Sec2 52


CovarianceandScatterPatterns


Figure513Jointprobabilitydistributionsandthesignofcov(X,Y).**Notethatcovarianceisameasureoflinearrelationship.**Variableswithnonzerocovariancearesaidtobecorrelated.

XandYareNOT

independent.

XandYareindependent.


Example520:IntuitiveCovariance


TheprobabilitydistributionofExample51isshown.

Byinspection,notethatthelargerprobabilitiesoccurasXandYmoveinoppositedirections.Thisindicatesanegativecovariance.

1 2 31 0.01 0.02 0.252 0.02 0.03 0.203 0.02 0.10 0.054 0.15 0.10 0.05

x=numberofbarsofsignalstrength

y=numberoftimescity

nameisstated

Y1

Y2

Y3Y4

0

0.05

0.1

0.15

0.2

0.25

12

3x

Y1

Y2

Y3

Y4


***Correlation( =rho)


The between random variables X and Y, denoted as , is

cov , (5-15)

Since 0 and 0, and cov , have the same sign.

We say that

correlation

XY

XYXY

X Y

X Y

XY

XY

X YV X V Y

X Y

is normalized, so 1 1 (5-16)Note that is dimensionless.Variables with non-zero correlation are correlated.

XY

XY


Example521:Covariance&CorrelationDeterminethe

covarianceandcorrelation.


x y f(x,y) xX yY Prod0 0 0.2 1.8 1.2 0.421 1 0.1 0.8 0.2 0.011 2 0.1 0.8 0.8 0.072 1 0.1 0.2 0.2 0.002 2 0.1 0.2 0.8 0.023 3 0.4 1.2 1.8 0.880 0.2 1.2601 0.2 0.9262 0.23 0.4

0 0.21 0.22 0.23 0.4

X = 1.8

Y = 1.8

X= 1.1662

Y= 1.1662

S

t

D

e

v

correlation=

J

o

i

n

t

M

a

r

g

i

n

a

l

covariance=

M

e

a

n

Notethestrongpositivecorrelation.

Figure514Discretejointdistribution,f(x,y).


Example521


Calculatethecorrelation.

Figure515Discretejointdistribution.Steepnessoflineconnectingpointsisimmaterial.


IfX andY areindependentrandomvariables,XY =XY =0 (517)

XY =0isnecessary,butnotasufficientcondition forindependence. Figure513d(x,y plotsasacircle)providesanexampleofdependence

DESPITE havingXY =XY =0. Figure513b(x,y plotsasasquare)providesanexampleof

independence AND havingXY =XY =0.

Ingeneral,arectangularpatterindicatesindependence,andthusXY =XY =0.

But,anonrectangularpatternwouldindicatedependence.

IndependenceImplies =0


Independent XY =XY =0 Dependentbut XY =XY =0 Dependentand XY ,XY 0


Example523:IndependenceImpliesZeroCovariance


Figure515Aplanarjointdistribution.

4 22 2

0 0

24 32

0 0

42

0

43

0

116

116 3

1 816 3

1 1 64 326 3 6 3 9

32 4 8 09 3 3

XY

E XY x y dx dy

xy dy

y dy

y

E XY E X E Y

4 22

0 0

24 3

0 0

42

0

4 22

0 0

24 22

0 0

43

0

116

116 3

1 8 1 16 416 2 3 6 2 3

116

116 2

2 1 64 816 3 8 3 3

E X x ydx dy

xy dy

y

E Y xy dx dy

xy dy

y

Let 16 for 0 2 and 0 4Show that 0

XY

XY

f xy x y x y

E XY E X E Y


Summary1:MarginalProbabilities Marginalprobabilitydistributionfunction

ExpectationandVariance

Sec2 61

Xy

Yx

f x f xy

f y f xy

,

,

X XYy

Y XYx

f x f x y dy

f y f x y dx


Summary2:CovarianceandCorrelation

Covariance

Correlation

IfX andY areindependentrandomvariables,XY =XY =0

TheoppositeisNOT true.

Sec2 62

joint probability distributions

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