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7. Two Random Variables

In many experiments, the observations are expressible not

as a single quantity, but as a family of quantities. For

example to record the height and weight of each person in

a community or the number of people and the total income

in a family, we need two numbers.

LetXand Ydenote two random variables (r.v) based on a

probability model (,F,P). Then

and

2

1

,)()()()( 1221x

xXXX dxxfxFxFxXxP

.)()()()(2

11221

y

y YYY dyyfyFyFyYyP PILLAI

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What about the probability that the pair of r.vs (X,Y) belongs

to an arbitrary regionD? In other words, how does one

estimate, for example,

Towards this, we define the joint probability distribution

function ofXand Yto be

wherex andy are arbitrary real numbers.

Properties

(i)

since we get

?))(())(( 2121 yYyxXxP

,0),(

))(())((),(

yYxXP

yYxXPyxFXY

(7-1)

.1),(,0),(),( XYXYXY FxFyF

,)()(,)( XyYX

(7-2)

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Similarly

we get

(ii)

To prove (7-3), we note that for

and the mutually exclusive property of the events on the

right side gives

which proves (7-3). Similarly (7-4) follows.

.0)(),( XPyFXY ,)(,)( YX

.1)(),( PFXY

).,(),()(,)( 1221 yxFyxFyYxXxP XYXY

).,(),()(,)( 1221 yxFyxFyYyxXP XYXY

(7-3)

(7-4)

,12

xx

yYxXxyYxXyYxX )(,)()(,)()(,)( 2112

yYxXxPyYxXPyYxXP )(,)()(,)()(,)( 2112

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(iii)

This is the probability that (X,Y) belongs to the rectangle

in Fig. 7.1. To prove (7-5), we can make use of the

following identity involving mutually exclusive events on

the right side.

).,(),(

),(),()(,)(

1121

12222121

yxFyxF

yxFyxFyYyxXxP

XYXY

XYXY

(7-5)

.)(,)()(,)()(,)( 2121121221 yYyxXxyYxXxyYxXx

0R

1y

2y

1x 2x

X

Y

Fig. 7.1

0R

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2121121221 )(,)()(,)()(,)( yYyxXxPyYxXxPyYxXxP

2yy 1y

.

),(),(

2

yx

yxFyxf XYXY

(7-6)

.),(),(

dudvvufyxFx y

XYXY (7-7)

.1),(

dxdyyxfXY (7-8)

This gives

and the desired result in (7-5) follows by making use of (7-3) with and respectively.

Joint probability density function (Joint p.d.f)

By definition, the joint p.d.f ofXand Yis given by

and hence we obtain the useful formula

Using (7-2), we also get

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To find the probability that (X,Y) belongs to an arbitrary

regionD, we can make use of (7-5) and (7-7). From (7-5)

and (7-7)

Thus the probability that (X,Y) belongs to a differentialrectangle xy equals and repeating this

procedure over the union of no overlapping differential

rectangles inD, we get the useful result

.),(),(

),(),(),(),()(,)(

yxyxfdudvvuf

yxFyxxFyyxFyyxxFyyYyxxXxP

XY

xx

x

yy

yXY

XYXYXY

XY

(7-9)

x

X

Y

Fig. 7.2

yD

,),( yxyxfXY

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Dyx XY dxdyyxfDYXP ),( .),(),( (7-10)

(iv) Marginal Statistics

In the context of several r.vs, the statistics of each individualones are called marginal statistics. Thus is the

marginal probability distribution function ofX, and is

the marginal p.d.f ofX. It is interesting to note that all

marginals can be obtained from the joint p.d.f. In fact

Also

To prove (7-11), we can make use of the identity

.),()(),,()( yFyFxFxF XYYXYX (7-11)

.),()(,),()(

dxyxfyfdyyxfxf XYYXYX (7-12)

)()()( YxXxX

)(xFX)(xfX

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so that

To prove (7-12), we can make use of (7-7) and (7-11),

which gives

and taking derivative with respect tox in (7-13), we get

At this point, it is useful to know the formula for

differentiation under integrals. Let

Then its derivative with respect tox is given by

Obvious use of (7-16) in (7-13) gives (7-14).

).,(,)( xFYxXPxXPxF XYX

dudyyufxFxFx

XYXYX ),(),()(

(7-13)

.),()(

dyyxfxf XYX (7-14)

.),()()(

)(

xb

xa

dyyxhxH (7-15)

( )

( )

( ) ( ) ( ) ( , )( , ) ( , ) .

b x

a x

dH x db x da x h x yh x b h x a dy

dx dx dx x

(7-16)

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IfXand Yare discrete r.vs, then represents

their joint p.d.f, and their respective marginal p.d.fs are

given by

and

Assuming that is written out in the form of a

rectangular array, to obtain from (7-17), one need to

add up all entries in the i-th row.

),(jiij

yYxXPp

j j

ijjii pyYxXPxXP ),()(

i i

ijjij pyYxXPyYP ),()(

(7-17)

(7-18)

),( ji yYxXP

),( ixXP

mnmjmm

inijii

nj

nj

pppp

pppp

pppppppp

21

21

222221

111211

j

ijp

i

ijp

Fig. 7.3

It used to be a practice for insurancecompanies routinely to scribble out

these sum values in the left and top

margins, thus suggesting the name

marginal densities! (Fig 7.3).

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From (7-11) and (7-12), the joint P.D.F and/or the joint p.d.f

represent complete information about the r.vs, and their

marginal p.d.fs can be evaluated from the joint p.d.f. However,

given marginals, (most often) it will not be possible tocompute the joint p.d.f. Consider the following example:

Example 7.1: Given

Obtain the marginal p.d.fs and

Solution: It is given that the joint p.d.f is a constant in

the shaded region in Fig. 7.4. We can use (7-8) to determinethat constant c. From (7-8)

.otherwise0,,10constant,),( yxyxfXY (7-19)

),( yxfXY

)(xfX ).(yfY

.122

),(

1

0

21

0

1

0

0

ccycydydydxcdxdyyxf

yy

y

xXY

(7-20)

0 1

1

X

Y

Fig. 7.4

y

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Thus c = 2. Moreover from (7-14)

and similarly

Clearly, in this case given and as in (7-21)-(7-22),it will not be possible to obtain the original joint p.d.f in (7-

19).

Example 7.2:Xand Yare said to be jointly normal (Gaussian)

distributed, if their joint p.d.f has the following form:

,10),1(22),()(1

xyXYX xxdydyyxfxf (7-21)

.10,22),()(

0

y

xXYY yydxdxyxfyf (7-22)

)(xfX )(yfY

.1||,,

,12

1),(

2

2

2

2

2

)(

))((2

)(

)1(2

1

2

yx

eyxf YY

YX

YX

X

X yyxx

YX

XY

(7-23)

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By direct integration, using (7-14) and completing the

square in (7-23), it can be shown that

~

and similarly

~

Following the above notation, we will denote (7-23)

as Once again, knowing the marginals

in (7-24) and (7-25) alone doesnt tell us everything about

the joint p.d.f in (7-23).

As we show below, the only situation where the marginal

p.d.fs can be used to recover the joint p.d.f is when the

random variables are statistically independent.

),,(2

1

),()(22/)(

2

22

XX

x

X

XYX NedyyxfxfXX

(7-24)

(7-25)),,(2

1),()( 2

2/)(

2

22

YY

y

Y

XYY NedxyxfyfYY

).,,,,(22 YXYXN

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Independence of r.vs

Definition: The random variablesXand Yare said to be

statistically independent if the events and

are independent events for any two Borel setsA andB inx

andy axes respectively. Applying the above definition to

the events and we conclude that, if

the r.vsXand Yare independent, then

i.e.,

or equivalently, ifXand Yare independent, then we must

have

AX )( })({ BY

xX )( ,)( yY

))(())(())(())(( yYPxXPyYxXP (7-26)

)()(),( yFxFyxF YXXY

).()(),( yfxfyxf YXXY (7-28)

(7-27)

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IfXand Yare discrete-type r.vs then their independence

implies

Equations (7-26)-(7-29) give us the procedure to test for

independence. Given obtain the marginal p.d.fs

and and examine whether (7-28) or (7-29) is valid. If

so, the r.vs are independent, otherwise they are dependent.

Returning back to Example 7.1, from (7-19)-(7-22), we

observe by direct verification that Hence

Xand Yare dependent r.vs in that case. It is easy to see thatsuch is the case in the case of Example 7.2 also, unless

In other words, two jointly Gaussian r.vs as in (7-23) are

independent if and only if the fifth parameter

.,allfor)()(),( jiyYPxXPyYxXPjiji

(7-29)

)(xfX

)(yfY

),,( yxfXY

).()(),( yfxfyxf YXXY

.0

.0

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Example 7.3: Given

Determine whetherXand Yare independent.

Solution:

Similarly

In this case

and henceXand Yare independent random variables.

otherwise.,0

,10,0,),(

2

xyexyyxf

y

XY (7-30)

.10,222

),()(

00

0

2

0

xxdyyeyex

dyeyxdyyxfxfyy

y

XYX

(7-31)

.0,2),()(

21

0 yey

dxyxfyf yXYY (7-32)

),()(),( yfxfyxf YXXY

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