251v2out 4/19/2006 (Open this document in 'Outline' view!)
K. Two Random Variables.
1. Regression (Summary).
2. Covariance (  xy and s xy )
a. Population Covariance
The population covariance is defined, using probability, as
Cov( x, y )   xy  E x   x  y   y  E xy    x  y . This can be used to describe the relationship



between x and y . If the covariance is positive we can say that x and y tend to move together, while if it
is negative we can say that they tend to move in opposite directions. In order to use this formula we must
realize that E xy    xyPx, y  . This means that we must add together the product of x and y ,
together with their joint probability, for each possible pair of values of x and y . For example, assume that
x
x and y are related by the following joint probability table:
400
y 600
800





400
.12
600
.15
.10
.05
.16
.07
800 .
.18 

.08 
.09 

We begin by taking the upper left hand probability, .12, which is the probability that both x and y are
400, and multiplying it by 400 twice. Then we take the next probability in the same row, .15, which is the
probability that x is 600 and y is 400, and multiply it by both 600 and 400. If we continue in this way we
get
 .12 400 400  .15600 400  .18800 400 
E xy  
xyPxy    .10 400 600  .05600 600  .08800 600 
  .16 400 800  .07 600 800  .09 800 800 

 19200 36000 57600 
  24000 18000 38400   335600 .
 51200 33600 57600 
We can now use the following tableau to compute the means and variances of x and y .
x
400
600
800
P y 
yP y  y 2 P y 
 .12
400
.15
.18 
.45
180
72000


y
600
.05
.08 
.23
138
82800
 .10
 .16
800
.07
.09 
.32
256 204800


Px 
.38
.27
.35
1.00 574 359600
xPx 
152 
162 
280 
594
2
x Px  60800  97200  224000  382000
 Px   1 (a check),   E x    xPx   594 , E x    x
 P y   1 ,   E y    yP y   574 and E y    y P y   359600
2
To summarize
x
2
2
Px   382000 ,
2
y
We will need the variances below. To complete what we have done, write
 xy  Covxy  Exy   x  y  335600 594574  5356
b. The Sample Covariance
The sample covariance is much easier to compute, the formula being
s xy 
 x  x  y  y    xy  nx y .
n 1
n 1
For example, assume that we have data on income ( x ) and savings ( y )(in thousands) for 5 families.
x
y
x2
y2
xy
1
2
3
4
5
1.9
12.4
6.4
7.0
7.0
0.0
0.9
0.4
1.2
0.3
3.61
153.76
40.96
49.00
49.00
0.00
0.81
0.16
1.44
0.09
0.00
11.16
2.56
8.40
2.10
Sum
34.7
2.8
296.33
2.50
24.22
Family
Then x 
s x2
34 .7
2 .8
 6.94 and y 
 0.56 .
5
5
x

2
 nx 2
n 1
y


296 .33  56.94 2
 13 .878 ,
4
2.50  50.56 2
 0.2330 and since
n 1
4
24.22  56.94 0.56 
xy  24 .22 , s xy 
 1.197 .
5 1
The positive sign of s xy , the sample covariance, indicates that x and y tend to move together.
s 2y
2
 ny 2


2
3. The Correlation Coefficient (  xy and rxy )
The size of a covariance is relatively meaningless; to judge the strength of the relationship between x and y
we need to compute the correlation, which is found by dividing the covariance by the standard deviations of
x and y.
a. Population Correlation.
For the population covariance, recall from above that
 
 x2  E x 2   x2  382000  594 2  29164 and
 
 y2  E y 2   y2  359600  5742  30124 . So that
 xy 
 xy
 x y

 5356

 5356
 0.181 .
170 .77 173 .56 
29164 30124
The correlation must always be between positive and negative 1  1.0    1.0 . A correlation close to
zero is called weak. A correlation that is close to one in absolute value is called strong. (Actually
statisticians prefer to look at the value of the correlation squared.) A strong positive correlation indicates
that x and y have a relationship that is close to a straight line with a positive slope. A strong negative
correlation means that the relationship approximates a straight line with a negative slope. Unfortunately, the
correlation only indicates linear relationships; a nonlinear relationship that is obvious on a graph may give a
zero correlation.
b. Sample Correlation.
Recall that s xy  1.197 , s x2  13 .878 , and s 2y  0.2330 . If we divide the
correlation by the two standard deviations, we find that rxy 
s xy
sx s y

1.197
 0.6657 .
13 .878 0.2330
4. Functions of Two Random Variables.
Cov(ax  b, cy  d )  acCov( x, y)
and if w  ax  b and v  cy  d ,  wv  signac xy or Corr (ax  b, cy  d )  (sign(ac))Corr ( x, y) ,
where signac has the value 1 or 1 depending on whether the product of a and c is negative or
positive.
5. Sums of Random Variables.
a. Ex  y   Ex  E y  and
Var x  y    x2   y2  2 xy  Var x   Var y   2Covx, y 
b. Independence.
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(i) Definition.
Px, y   Px P y 
(ii) Consequences
If x and y are independent,
E xy   E x E  y  , Covx, y   0 ,  xy  0
and Varx  y   Varx   Var y  .
c. If a, c and d are constants, Var(ax  cy)  a 2Var( x)  c 2Var( y)  2acCov( x, y) . This and a. imply that
Eax  cy  d   aEx   cE y   d
and Var(ax  cy  d )  a 2Var( x)  c 2Var( y)  2acCov( x, y)
d. Application to portfolio analysis
If R  P1 R1  P2 R2 and P1  P2  1 , then E R   P1 E R1   P2 E R2  and
VarR  P12VarR1   P22VarR2   2P1 P2CovR1 , R2  . Variance is usually considered a measure of risk,
though actually, the best measure of risk is probably the coefficient of variation, the standard deviation

divided by the mean, in this case C  R .
E R 
The remainder of this material can be found in the Supplement in the document 251var2.
You can get a slightly expanded version of this at 251varmin .
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