The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

Appendix 4A Exercises
4.1. “If two random variables are statistically independent, the coefficient of correlation between the
two is zero. But the converse is not necessarily true; that is, zero correlation does not imply
statistical independence. However, if two variables are normally distributed, zero correlation
necessarily implies statistical independence.” Verify this statement for the following joint
probability density function of two normally distributed variables 
Y
1
and 
Y
2
(this joint
probability density function is known as the 
bivariate normal probability density function
):
f
(
Y
1
,
Y
2
)
=
1
2
πσ
1
σ
2
1
−
ρ
2
exp
−
1
2(1
−
ρ
2
)
×
Y
1
−
µ
1
σ
1
2
−
2
ρ
(
Y
1
−
µ
1
)(
Y
2
−
µ
2
)
σ
1
σ
2
+
Y
2
−
µ
2
σ
2
2
4
See 
Appendix A
for a general discussion of the properties of the maximum likelihood estimators as
well as for the distinction between asymptotic unbiasedness and consistency. Roughly speaking, in
asymptotic unbiasedness we try to find out the lim
E
(
˜
σ
2
n
) as 
n
tends to infinity, where 
n
is the sample
size on which the estimator is based, whereas in consistency we try to find out how 
˜
σ
2
n
behaves as 
n
increases indefinitely. Notice that the unbiasedness property is a repeated sampling property of an
estimator based on a sample of given size, whereas in consistency we are concerned with the
behavior of an estimator as the sample size increases indefinitely.
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