Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

126 Multivariate Distributions

It follows by the linearity of expectation, Theorem 2.1.1, that the covariance of
XandY can also be expressed as

cov(X, Y)=E(XY−μ 2 X−μ 1 Y+μ 1 μ 2 )

= E(XY)−μ 2 E(X)−μ 1 E(Y)+μ 1 μ 2

= E(XY)−μ 1 μ 2 , (2.5.2)

which is often easier to compute than using the definition, (2.5.1).

The measure that we seek is a standardized (unitless) version of the covariance.

Definition 2.5.2.If each ofσ 1 andσ 2 is positive, then thecorrelation coefficient

betweenXandYis defined by

ρ=

E[(X−μ 1 )(Y−μ 2 )]

σ 1 σ 2

=

cov(X, Y)

σ 1 σ 2

. (2.5.3)

It should be noted that the expected value of the product of two random variables
is equal to the product of their expectations plus their covariance; that is,E(XY)=
μ 1 μ 2 +cov(X, Y)=μ 1 μ 2 +ρσ 1 σ 2.
As illustrations, we present two examples. The first is for a discrete model while
the second concerns a continuous model.

Example 2.5.1.Reconsider the random vector (X 1 ,X 2 ) of Example 2.1.1 where a
fair coin is flipped three times andX 1 is the number of heads on the first two flips
whileX 2 is the number of heads on all three flips. Recall that Table 2.1.1 contains
the marginal distributions ofX 1 andX 2. By symmetry of these pmfs, we have
E(X 1 )=1andE(X 2 )=3/2. To compute the correlation coefficient of (X 1 ,X 2 ),
we next sketch the computation of the required moments:

E(X^21 )=

1

2

+2^2 ·

1

4

=

3

2

⇒σ^21 =

3

2

− 12 =

1

2

;

E(X^22 )=

3

8

+4·

3

8

+9·

1

8

=3⇒σ^22 =3−

(

3

2

) 2

12 =

1

2

;

E(X 1 X 2 )=
2
8

+1· 2 ·

2

8

+2· 2 ·

1

8

+2· 3 ·

1

8

=2⇒cov(X 1 ,X 2 )=2− 1 ·

3

2

=

1

2

From which it follows thatρ=(1/2)/(

√

(1/2)

√

3 /4) = 0.816.

Example 2.5.2.Let the random variablesXandYhave the joint pdf

f(x, y)=

{

x+y 0 <x< 1 , 0 <y< 1

0elsewhere.

We next compute the correlation coefficientρofXandY.Now

μ 1 =E(X)=

∫ 1

0

∫ 1

0

x(x+y)dxdy=

7

12