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Brigham Young University Department of Economics
Economics 459  International Monetary Theory
Useful Properties of Expected Values, Variances & Covariances
The expected value, or mean, of a random variable, say
x
, is defined by,
∫
=
dx
x
f
x
x
E
)
(
}
{
, where
f
(
x
) is the probability density function.
The variance of a random variables is defined as,
}
})
{
{(
}
{
2
x
E
x
E
x
Var
−
=
, and the
square root of this is called the standard deviation.
The standard deviation can be
interpreted as how far from the mean the variable is on average.
Both the standard
deviation and the variance must be positive.
The covariance of two random variables, say
x
1
&
x
2
, is defined as,
})}
{
})(
{
{(
}
,
{
2
2
1
1
2
1
x
E
x
x
E
x
E
x
x
Cov
−
−
=
.
Note that if
x
1
tends to be above its mean
when
x
2
is above its mean also, this will evaluate to a positive value.
If
x
1
tends to be
below its means when
x
2
is above its, then the covariance will be negative.
Two random
variables are said to be “independent” if their covariance is zero.
Also note that if
x
2
=
x
1
,
i.e. we consider the covariance of
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 Winter '08
 Phillips,K
 Economics

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