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10. Joint Moments and Joint Characteristic Functions

Following section 6, in this section we shall introduce various parameters to compactly represent the information contained in the joint p.d.f of two r.vs. Given two r.vs X and Y and a function define the r.v

Using (6-2), we can define the mean of Z to be

(10-1)),( YXgZ

),,( yxg

.)( )(

dzzfzZE ZZ (10-2)

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However, the situation here is similar to that in (6-13), and it is possible to express the mean of in terms of without computing To see this, recall from (5-26) and (7-10) that

where is the region in xy plane satisfying the above inequality. From (10-3), we get

As covers the entire z axis, the corresponding regions are nonoverlapping, and they cover the entire xy plane.

zDyxXY

Z

yxyxf

zzYXgzPzzfzzZzP

),(

),(

),()(

(10-3)

zD

),( YXgZ ),( yxf XY ).(zfZ

( , )

( ) ( , ) ( , ) .z

Z XYx y D

z f z z g x y f x y x y

(10-4)

zDz

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By integrating (10-4), we obtain the useful formula

or

If X and Y are discrete-type r.vs, then

Since expectation is a linear operator, we also get

( ) ( ) ( , ) ( , ) .Z XYE Z z f z dz g x y f x y dxdy

(10-5)

(10-6)

[ ( , )] ( , ) ( , ) .XYE g X Y g x y f x y dxdy

[ ( , )] ( , ) ( , ).i j i ji j

E g X Y g x y P X x Y y (10-7)

( , ) [ ( , )].k k k kk k

E a g X Y a E g X Y (10-8)

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If X and Y are independent r.vs, it is easy to see that and are always independent of each other. In that case using (10-7), we get the interesting result

However (10-9) is in general not true (if X and Y are not independent).

In the case of one random variable (see (10- 6)), we defined the parameters mean and variance to represent its average behavior. How does one parametrically represent similar cross-behavior between two random variables? Towards this, we can generalize the variance definition given in (6-16) as shown below:

)(XgZ

)].([)]([)()()()(

)()()()()]()([

YhEXgEdyyfyhdxxfxg

dxdyyfxfyhxgYhXgE

YX

YX

(10-9)

)(YhW

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Covariance: Given any two r.vs X and Y, define

By expanding and simplifying the right side of (10-10), we also get

It is easy to see that

To see (10-12), let so that

.

)()()()(),(________

YXXY

YEXEXYEXYEYXCov YX

(10-10)

(10-12)

. )()(),( YX YXEYXCov

(10-11)

,YaXU

. )()(),( YVarXVarYXCov

. 0)(),( 2)(

)()()(2

2

YVarYXCovaXVara

YXaEUVar YX (10-13)

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The right side of (10-13) represents a quadratic in the variable a that has no distinct real roots (Fig. 10.1). Thus the roots are imaginary (or double) and hence the discriminant

must be non-positive, and that gives (10-12). Using (10-12), we may define the normalized parameter

or

and it represents the correlation coefficient between X and Y.

(10-14)

)( )(),( 2 YVarXVarYXCov

,11 ,),(

)()(

),( XY

YXXY

YXCov

YVarXVar

YXCov

a

)(UVar

Fig. 10.1

YXXYYXCov ),( (10-15)

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Uncorrelated r.vs: If then X and Y are said to be uncorrelated r.vs. From (11), if X and Y are uncorrelated, then

Orthogonality: X and Y are said to be orthogonal if

From (10-16) - (10-17), if either X or Y has zero mean, then orthogonality implies uncorrelatedness also and vice-versa. Suppose X and Y are independent r.vs. Then from (10-9) with we get

and together with (10-16), we conclude that the random variables are uncorrelated, thus justifying the original definition in (10-10). Thus independence implies uncorrelatedness.

(10-16)

,0XY

,)( ,)( YYhXXg

).()()( YEXEXYE

.0)( XYE(10-17)

),()()( YEXEXYE

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Naturally, if two random variables are statistically independent, then there cannot be any correlation between them However, the converse is in general not true. As the next example shows, random variables can be uncorrelated without being independent.

Example 10.1: Let Suppose X and Y are independent. Define Z = X + Y, W = X - Y . Show that Z and W are dependent, but uncorrelated r.vs.

Solution: gives the only solution set to be

Moreover and

),1,0( UX ).1,0( UY

|| ,2 ,2 ,11 ,20 wzwzwzwz .2/1|),(| wzJ

.2

,2

wzy

wzx

yxwyxz ,

).0( XY

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,otherwise,0

,|| ,2 ,2 ,11 ,20,2/1),(

zwwzwzwzwzfZW

(10-18)

Thus (see the shaded region in Fig. 10.2)

and hence

or by direct computation ( Z = X + Y )

Fig. 10.21

1

w

z2

,21,2 2

1

,10 , 2

1

),()(2

2

zzdw

zzdw

dwwzfzf-z

z-

z

z

ZWZ

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and

Clearly Thus Z and W are not independent. However

and

and hence

implying that Z and W are uncorrelated random variables.

,otherwise,0

,21,2

,10,

)()()( zz

zz

zfzfzf YXZ(10-19)

(10-20)

).()(),( wfzfwzf WZZW

(10-21) ,0)()())(()( 22 YEXEYXYXEZWE

0)()()(),( WEZEZWEWZCov (10-22)

.otherwise,0

,11|,|1

2

1),()(

||2

wwdzdzwzfwf

w

|w|ZWW

,0)()( YXEWE

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Example 10.2: Let Determine the variance of Z in terms of and Solution:

and using (10-15)

In particular if X and Y are independent, then and (10-23) reduces to

Thus the variance of the sum of independent r.vs is the sum of their variances

(10-23)

,0XY

.bYaXZ YX , .XY

( ) ( )Z X YE Z E aX bY a b

22 2

2 2 2 2

2 2 2 2

( ) ( ) ( ) ( )

( ) 2 ( )( ) ( )

2 .

Z Z X Y

X X Y Y

X XY X Y Y

Var Z E Z E a X b Y

a E X abE X Y b E Y

a ab b

.22222YXZ ba (10-24)

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Moments:

represents the joint moment of order (k,m) for X and Y.

Following the one random variable case, we can define the joint characteristic function between two random variables which will turn out to be useful for moment calculations.

Joint characteristic functions:

The joint characteristic function between X and Y is defined as

Note that

, ),(][

dydxyxfyxYXE XY

mkmk (10-25)

( ) ( )

( , ) ( , ) .j Xu Yv j Xu Yv

XY XYu v E e e f x y dxdy

(10-26)

.1)0,0(),( XYXY vu

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It is easy to show that

If X and Y are independent r.vs, then from (10-26), we obtain

Also

More on Gaussian r.vs :

From Lecture 7, X and Y are said to be jointly Gaussian as if their joint p.d.f has the form in (7-23). In that case, by direct substitution and simplification, we obtain the joint characteristic function of two jointly Gaussian r.vs to be

.),(1

)(0,0

2

2

vu

XY

vu

vu

jXYE (10-27)

).()()()(),( vueEeEvu YXjvYjuX

XY (10-28)

( ) ( , 0), ( ) (0, ).X XY Y XYu u v v (10-29)

),,,,,( 22 YXYXN

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.)(),()2(

2

1)()(

2222 vuvuvujYvXujXY

YYXXYX

eeEvu (10-30)

Equation (10-14) can be used to make various conclusions. Letting in (10-30), we get

and it agrees with (6-47).

From (7-23) by direct computation using (10-11), it is easy to show that for two jointly Gaussian random variables

Hence from (10-14), in represents the actual correlation coefficient of the two jointly Gaussian r.vs in (7-23). Notice that implies

0v

,)0,()(22

2

1uuj

XYX

XX

euu

(10-31)

),,,,( 22 YXYXN

. ),( YXYXCov

0

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Thus if X and Y are jointly Gaussian, uncorrelatedness does imply independence between the two random variables. Gaussian case is the only exception where the two concepts imply each other.

Example 10.3: Let X and Y be jointly Gaussian r.vs with parameters Define Determine Solution: In this case we can make use of characteristic function to solve this problem.

).,,,,( 22 YXYXN .bYaXZ ).(zfZ

).,(

)()()()( )(

buau

eEeEeEu

XY

jbuYjauXubYaXjjZuZ

(10-32)

).()(),( yfxfYXf YXXY

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From (10-30) with u and v replaced by au and bu respectively we get

where

Notice that (10-33) has the same form as (10-31), and hence we conclude that is also Gaussian with mean and variance as in (10-34) - (10-35), which also agrees with (10-23).

From the previous example, we conclude that any linear combination of jointly Gaussian r.vs generate a Gaussian r.v.

,)(2222222

2

1)2(

2

1)( uujubabaubaj

Z

ZZYYXXYX

eeu

(10-33)

(10-34)

(10-35)

bYaXZ

.2

,22222YYXXZ

YXZ

baba

ba

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In other words, linearity preserves Gaussianity. We can use the characteristic function relation to conclude an even more general result.

Example 10.4: Suppose X and Y are jointly Gaussian r.vs as in the previous example. Define two linear combinations

what can we say about their joint distribution? Solution: The characteristic function of Z and W is given by

As before substituting (10-30) into (10-37) with u and v replaced by au + cv and bu + dv respectively, we get

. , dYcXWbYaXZ (10-36)

).,()(

)()(),()()(

)()()(

dvbucvaueE

eEeEvu

XYdvbujYcvaujX

vdYcXjubYaXjWvZujZW

(10-37)

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,),()2(

2

1)( 2222 vuvuvuj

ZW

WYXZWZWZ

evu

(10-38)

where

and

From (10-38), we conclude that Z and W are also jointly distributed Gaussian r.vs with means, variances and correlation coefficient as in (10-39) - (10-43).

,2

,2

,

,

22222

22222

YYXXW

YYXXZ

YXW

YXZ

dcdc

baba

dc

ba

(10-39)

(10-40)

(10-41)

(10-42)

.)( 22

WZ

YYXXZW

bdbcadac

(10-43)

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To summarize, any two linear combinations of jointly Gaussian random variables (independent or dependent) are also jointly Gaussian r.vs.

Of course, we could have reached the same conclusion by deriving the joint p.d.f using the technique developed in section 9 (refer (7-29)).

Gaussian random variables are also interesting because of the following result:

Central Limit Theorem: Suppose are a set of zero mean independent, identically distributed (i.i.d) random

Linear operator

Gaussian input Gaussian output

),( wzfZW

nXXX ,,, 21

Fig. 10.3

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variables with some common distribution. Consider their scaled sum

Then asymptotically (as )

Proof: Although the theorem is true under even more general conditions, we shall prove it here under the independence assumption. Let represent their common variance. Since

we have

.21

n

XXXY n

(10-44)

n

).,0( 2NY

2

,0)( iXE

.)()( 22 ii XEXVar

(10-45)

(10-46)

(10-47)

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Consider

where we have made use of the independence of the r.vs But

where we have made use of (10-46) - (10-47). Substituting (10-49) into (10-48), we obtain

and as

n

iX

n

i

nujXnuXXXjjYuY

nu

eEeEeEu

i

in

1

1

//)(

)/(

)()()( 21

(10-48)

.,,, 21 nXXX

,1

21

!3!21)(

2/3

22

2/3

333222/

no

n

u

n

uXj

n

uXj

n

ujXEeE iiinujX i

(10-49)

,1

21)(

2/3

22 n

Y no

n

uu

(10-50)

2 2 / 2lim ( ) ,uY

nu e

(10-51)

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.1lim xn

ne

n

x

(10-52)

[Note that terms in (10-50) decay faster than But (10-51) represents the characteristic function of a zero mean normal r.v with variance and (10-45) follows.

The central limit theorem states that a large sum of independent random variables each with finite variance tends to behave like a normal random variable. Thus the individual p.d.fs become unimportant to analyze the collective sum behavior. If we model the noise phenomenon as the sum of a large number of independent random variables (eg: electron motion in resistor components), then this theorem allows us to conclude that noise behaves like a Gaussian r.v.

3 / 2(1/ )o n 3 / 21/ ].n

2

since

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It may be remarked that the finite variance assumption is necessary for the theorem to hold good. To prove its importance, consider the r.vs to be Cauchy distributed, and let

where each Then since

substituting this into (10-48), we get

which shows that Y is still Cauchy with parameter In other words, central limit theorem doesn’t hold good for a set of Cauchy r.vs as their variances are undefined.

.21

n

XXXY n

(10-53)

).( CX i

,)( ||uX eu

i

(10-54)

n| |/

1

( ) ( / ) ~ ( ),n

u nY X

i

u u n e C n

(10-55)

.n

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Joint characteristic functions are useful in determining the p.d.f of linear combinations of r.vs. For example, with X and Y as independent Poisson r.vs with parameters and respectively, let

Then

But from (6-33)

so that

i.e., sum of independent Poisson r.vs is also a Poisson random variable.

.YXZ (10-56)

).()()( uuu YXZ (10-57)

)1()1( 21 )( ,)( juju e

Ye

X eueu (10-58)

)( )( 21)1)(( 21 Peu

jueZ

(10-59)

1 2

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