Suppose
X
has a symmetric distribution about zero, and let
Y
X
2
.
Then
E
Y

X
X
2
≠
E
Y
Var
X
(a number). But
Cov
X
,
Y
E
X
3
0.
∙
At this point, we can summarize what we know about the three kinds
of dependence we have discussed:
X
and
Y
independent
E
Y

X
E
Y
E
Y

X
E
Y
Cov
X
,
Y
0
But neither of the reverse implications holds.
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