More on Random Variables
Expectation :: Properties :: Pairs :: Independence :: Two R.V.S. :: Functions :: Inequalities :: Conditional :: General
Expectations of functions of two r.v.s
Let
be measurable (e.g.,
). Then for a
bivariate r.v.
we can define
.
Properties:
 If then .

If
and
are independent then
However, if for all appropriate functions, then and are independent. In fact, this is necessary and sufficient for independence.
 . .

If
and
are independent then
. If
, we say that
and
are
uncorrelated.
Again, uncorrelated does not imply independence.
 . If then .

for all constants
. Thus

. This can be shown using the CauchySchwartz
inequality.
iff and are linearly related,
As we have observed before, if are jointly Gaussian and , then they are independent. Otherwise, does not imply independence.