8/18/2019 bivarnorm

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5.12

The Bivariate

Normal

Distribution

313

5

T

he iv

ariat

e

 

orm

 l

 ist

ribu

tion

The

first

multivariate

continuous

distribution

for

which

we have

a

name is

a

generalization of

the

normal

distribution

to two

coordinates.

There is more

structure

to the bivanate

normal distribution

than just

a pair

of

normal

marginal

distributions.

Definition

of t

 

Bivarlate

Normal Distribution

Suppose

that Z

and

Z

are independent

random

variables,

each

of

which

has

a

standard

normal

distribution.

Then the

joint

p.d.f.

2

of Z 

and Z is specified

for all

values

of

and

z

by th e

equation

g

 zz, =

  exp[—— z

+

z

].

 5.12.1

2

L

2

For

constants

 

/12

a

. a,,

and

p

such that —00

0

 i

= 1,

2 .

and —1

<

p

<

1 we

shall

now define

two new

random variables

X

and X,

as

follows:

X =a

Z

-i—i

X,

= a

[pZi

+

 1  p

2z

+

it .

 5.12.2

We

shall

derive

the joint

p.d.f. f

 x

  X2

of

X and X,.

  he

transformation

from

Z

  and

1,

to

X and X is

a linear transformation;

and

it

will be found

that the determinant

of

the

matrix

of

coefficients

of

Z 

and

Z2

has

the

value

z\ =  1

 

p 

2

 

Therefore,

as

discussed

in

Section

3.9, the

Jacobian

J

of the

inverse transformation

from

X

and K,

to Z  and

Z2 is

J

=   =

.

 5.12.3

l—p- 

aa2

Since

J

>

0.

the

value

of

IJI

is equal

to

the

value

of

J

itself. If the

relations

 5.12.2

are solved for

Zi

and

Z

2

in terms of X

and

X

then the

joint p.d.f. f

 x

 

x-,

can

be

obtained

by

replacing

and

z

in

Eq .

 5.12.1

by their

expressions

in

terms ofx

and 52.

and then

multiplying

by

JI.

It

can

be shown

that

the

result

is, for

 

8/18/2019 bivarnorm

2/6

3

14

C

ha

pt

er

5 S

p

ec

ial

Di

str

ib

ut

io

ns

va

ria

nc

e

 

it

fo

ll

ow

s t

ha

t

E

 

X

 

=

 

E

 

X

 

p 

a

r

X

 

=

r

 

an

d

V

a

r

 X

j

F

u

rth

er

m

or

e

it

c

an

be s

h

ow

n

b

y u

sin

g

E

q

.  5

.1

2.

2

t

ha

t C

ov

 X

  X

 

P

lo

’ Th

er

ef

o

the

c

o

rre

la

tio

n

of X

an

d

X

:

is

si

m

pl

y

p

.

In

su

m

m

ary

.

if

X

a

nd

X

h

av

e

a

b

iva

ri

ate

no

rn

j

d

ist

hb

ut

io

n

fo

r w

h

ich

th

e

p

.d

.f

.

is

sp

ec

ifi

ed

b

y

E

q.

 

5.

12

.4

 ,

th

en

E

 X

i 

=j

.L

:

a

nd

V

a

r

 X

 

a

fo

r

i=

l

,2

.

A

ls

o,

p

 

X

 

X 

p.

It

ha

s

be

en

co

nv

en

ie

nt

fo

r u

s

to

in

tro

d

uc

e

th

e

bi

va

ria

te

n

or

m

al

di

str

ib

ut

ion

a

s

th

e

jo

in

t

di

st

rib

ut

io

n

of

c

ert

ai

n li

ne

ar

co

m

bin

at

io

ns of

ind

ep

e

nd

en

t r

an

do

m

va

ri

ab

le

s ha

v

i

ng sta

nd

ar

d

n

or

ma

l d

is

tri

bu

tio

ns

. I

t

sh

ou

ld

be

em

p

ha

siz

ed

 

h

ow

e

ve

r,

tha

t

th

e

b

iv

ar

iat

e

n

or

ma

l

d

ist

rib

ut

io

n

a

ris

es

dir

ec

tly

a

nd

n

at

ur

all

y

i

n m

a

ny pr

ac

tic

al

pr

ob

le

m

s.

F

or

e

xa

m

p

le,

fo

r m

a

ny

p

op

ul

at

ion

s

th

e jo

in

t d

is

tri

bu

ti

on

of

tw

o

p

hy

si

ca

l c

ha

ra

cte

ri

sti

cs

su

ch

a

s

th

e h

ei

gh

ts

an

d

th

e w

ei

gh

ts

o

f t

he

ind

iv

id

ua

ls

in th

e

po

pu

la

tio

n

w

il

l

b

e a

p

pro

x

im

at

ely

a

b

iva

ri

at

e n

or

m

al

d

ist

rib

ut

io

n.

Fo

r

oth

er

p

op

ul

at

ion

s,

t

he jo

in

t di

st

rib

ut

io

n

of th

e

s

co

re

s

of

th

e in

di

vi

du

al

s

in

th

e p

op

ul

ati

on

on

tw

o

re

la

ted

t

es

ts w

i

ll

b

e a

pp

ro

xi

ma

te

ly a b

iv

ari

at

e

n

or

m

al

di

str

ib

ut

ion

.

E

xa

m

pl

e

5.

12

.1 A

n

thr

op

o

me

tr

y

o

f F

le

a

Be

et

les

 

L

ub

isc

he

w

 1

96

2)

re

po

rts

th

e me

a

su

rem

e

nt

s

of s

ev

e

ra

l ph

y

sic

al

fe

atu

re

s o

f

a

va

rie

ty

of

s

pe

ci

es

of

f

lea b

ee

tl

e.

Th

e

in

ve

st

iga

ti

on

wa

s

c

on

ce

rn

ed

wi

th

w

he

th

er

s

om

e

c

o

mb

in

at

io

n

o

f ea

si

ly

o

bt

ai

ne

d

m

ea

su

rem

e

nt

s

c

ou

ld be

u

se

d

to

di

st

ing

u

ish

the

di

ff

ere

nt

s

pe

cie

s.

Fi

gu

re

5

.8

s

ho

w

s

a

sc

at

ter

pl

ot

o

f

m

e

as

ur

em

e

nt

s

of

th

e f

irs

t jo

in

t

in

th

e

f

irs

t

ta

rsu

s

ve

rsu

s

the s

ec

on

d jo

in

t

i

n t

he fi

rs

t ta

rsu

s f

or

a s

am

p

le

of

 

f

ro

m

th

e sp

ec

ie

s

Ch

ae

to

cn

em

a

h

ei

ke

rti

ng

er

i.

T

he

p

lo

t

als

o

in

clu

d

es

th

re

e

ell

ip

se

s

th

at

c

orr

es

po

n

d t

o a

fi

tte

d b

iv

ar

iat

e n

o

rm

al

di

str

ib

uti

on

.

T

h

e

ell

ip

se

s

w

e

re

c

h

os

en

t

o

c

on

ta

in

2

5 

.

50

 

.

a

nd

7

5 

of

the

p

ro

b

ab

ili

ty

o

f th

e

fit

te

d

b

iv

ar

ia

te

no

rm

a

l

dis

tr

ibu

ti

on

.

T

h

e

co

rre

la

tio

n o

f

the

fit

ted di

st

rib

ut

io

n

is

0

.6

4.

4

M

a

rg

in

a

l

a

n

d  

on

d

i

ti

on

a

l  

is

tr

ib

u

ti

o

n

s

M

arg

in

ai

fli

st

rib

ut

io

ns

 

W

e

s

ha

ll c

on

tin

u

e

to

a

ssu

m

e

th

at

th

e ra

nd

om

v

ar

ia

bl

es

X

a

nd

X

-, ha

ve

a

bi

va

ria

te

no

rm

a

l d

ist

rib

ut

io

n,

an

d th

ei

r

j

oin

t

p

.d.

f. is sp

ec

ifi

ed

by

Eq

.

 5

.1

2.

41

in

th

e

stu

d

y

o

f

the p

ro

p

ert

ie

s

o

f t

hi

s

d

is

tri

bu

tio

n

, it

w

ill

be

co

nv

en

ie

nt

to

re

pr

es

en

t

X

ar

id

X

a

s in

E

q.

 5

.12

.2

).

w

he

re

Z

an

d

Z

ar

e

ind

e

pe

nd

en

t r

an

do

m

va

ria

b

les

w

it

h

s

tan

d

ar

d

n

o

rm

al

di

str

ib

uti

on

s.

I

n

p

ar

tic

ul

ar .

sin

ce

b

ot

h

X

a

nd

X,

a

re

l

in

ea

r

co

m

bi

na

tio

n

s

of

I

a

nd

Z

. it

fo

ll

ow

s

fr

om

th

is r

ep

re

se

nta

ti

on

a

nd

fro

m

C

o

ro

lla

ry

5.

6.

1

th

at

th

e

m

a

r

dis

tr

ibu

ti

on

s of

bo

th

X

a

n

d

X

a

re

a

ls

o n

or

m

al d

is

tri

bu

ti

on

s.

Th

us

.

f

or

i

=

1

,2

,

th

e

m

arg

in

al d

is

tri

bu

tio

n

o

f

X

is

a

no

rm

a

l

di

str

ib

uti

on

w

ith

m

ea

n

 

1

an

d

v

ar

ia

nc

e

a

Independence

and

correlation 

If

X

and

X 

are

unco rrelated 

then p

=0  In

this

case

it

c

an

he

se

en fro

m

E

q.

 5

.1

2.

4)

th

at

t

he

jo

int

p.

d.f

.

j

x

.

x

’

fac

to

rs

in

to

the

p

r

o

du

c

t0

m

a

rg

in

al

p 

d f

of X

a

nd

th

e

ma

rg

in

al

p.

d.

f. o

f

X

H

e

nc

e, X

an

d

X ar

e

in

de

P

efl

dd

1

a

nd th

e fo

llo

w

in

g

re

su

lt h

a

s

be

en

e

sta

bl

ish

e

d:

8/18/2019 bivarnorm

3/6

 

IC   lVcflIO

IMUIIIICII

L. U IIIUULIUH

ii

t

Lu Junit

Figure

5.8

S

it rplut

tied  tt

Lita

iL

5

O dud 5’

hodridie

nuruta

e 

Iipe

Lu

E\urnple

5.

 .

I.

Tao

random

variables

.V

and

X

that hake

a

baariate

normal

distribution

are

independent

if

and

only if

they are

uncorrelated.

\Ve

have

already seen

in Section 4.6

that

tso

random

sariables X and

X v

ith

an

arbitrary joint

distribution

can

he

unconelated

ithout

being

independent.

 ‘o

ndit

iona

l

Dist

ribu

tions

.

The

conditional

distribution

of

X-. izie

n that

‘i

 

can

also

he

derived

from

the

representation

in Eq.

15.12.2). If

X  

.

then

Z1

 

—

 • Therefore,

the

conditional

distribution of

,V

e

ien that

.V

 

i

is the

same

as

the

conditional distribution

of

 I

2 t

2

 

+

 

L

+

 i

 5.12.5)

at

I

Because

Z has a standard

normal

distribution

and

is independent

of V it follows

from

5. 12.5

that

the

conditional

distribution of

.V

cisen

that

X

 

r

is a

normal

distribution.

for

which

the mean is

 

E

 Xx

 

p + pa

 

_/t

t

5.12.6)

and

the variance

is

 1 — 

a

 

The

conditional

distribution of

X

given

that

X

 

x

cannot he

deri’ed

so

easily

from

Eq.

5. 12.2 because

of

the different ways

in

s

hich

Z and

Z enter

Eq.

t

5. 12.2).

f o

seer

 

it is seen

from

Eq .

 5.12.4)

that the

Joint p d f

fL y  v is

symmetric

in

the

 

o ariables

 

— p

 

o

and

 v /L

 

nH

  Therefore, it

foIlos

s ihat

the

conditional

distribution

of X

gi en that

X

 

can he

found

from

the

conditional

distribution

of

  uRen

that

.V

 

this

distribLttion

has

Just

been deri’.ed)

simply

by

interchanging

5))

1

2i

.: ii J

ui

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5.1.2

The

Bivariate

Normal Distribution

317

 V

and

 

must

he

a

hi

ariate

nirmal

distribution see

Exercise 14>. Hence,

the

marginal

distribution

of

 

is

also a normal

disinbution.

[The

mean

and

the

ariance

of

 

niust

he

determined.

ihe mean

at

 

is

EiY

EIEiYXiJ

EX

—jr.

Eui

them

more.

h

E\ercmse

II

of

Section

4.7,

a

r Y>

E[Var

Y X

ii

Var[

Ei

Y

Xi

Er

—Var>X>

Hence,

he

distribution of   is

a normal

distribution

s oh mean  

and

variance r-

o

Linear

Combinations

.4

Suppose

again

that

two

random

variables X

and

X

hake

a

hivariate

normal distribution,

tar shich

the

p.d.f.

is

specified

by Eq.

5.12.4 .

Now

consider

the

random variable

Y

 

a

X

+

a

X

+

/

v

here

 

a

 

and

h

are arbitrary

given constants. Both X and

X

can he

represented.

as

in Eq.

 5.12.2 .

as linear

combinations

of

independent

and

normally

distributed

random variables

Z

and

Z 

Since

Y

is

a

linear

combination

of

X

and

X.

it follows

that V

c an a lso

be represented as

a

linear

combination

of

Z and Z

 

Therefore.

by

Corollary

56.1. the

distribution

of

V

will also

be a

normal

distribution.

Thus,

the

following

important

property

has

been established.

If

two random

variables

X

and

 V have a

b/variate

normal

distribution, then

each

linear

combination V

=

 

i

a 

+

h

will have a

normal distribution.

The

mean

and

variance

of

V are as

follows:

and

E>Y>

 

X 

+

 

X

 

+

b

‘il

i

+a2M2

 1

Van

Y

 

02

Var

K

 

a VariX

  2a

Cov>X

 

K 

c rT aa

 

±

2aap

on 5.12.8

Example

5.12.4

Heights

of Husbands

and Wives.

Suppose

that

a

maiTied couple

is selected

at

random

from

a

certain

population of

married couples.

and

that

the joint distribution of

the

height

of the

s

ife

and

the

height

of her

husband is a

bivariate

normal

distribution.

Suppose

t ha t t he

heights at

the wive s have

a

mean

of

66.8

inches and

a

standard deviation of

2

inches.

the

heights

of the

husbands

have

a

mean of 70 inches and

a

standard

deviation of

2

inches.

and

the

correlation

heteen

these t

o heights is

0.68.

We shall

determine

the

probability that

the

wife

will

he

taller than

he r

husband.

k

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