Lemma: Cauchy-Schwarz Inequality in the context of expectation.
Let
and
be two random variables such that ( ), ( ), and (
, (
)-
(
) (
) exist. Then
)
Pf.
Consider a new random variable
.
,(
(
(
If we now let
)
)
) -
(
)
(
)
(
)
and plug into the above inequality, we get
(
, (
(
)
))
, ( )( )
(
, ( )( )
)
This is just what we wanted to prove. ◘
Theorem. “Correlation is bounded by 1.”
Let
and
be two random variables such that ( ), ( ), and (
|
(
) exist. Then
)|
Pf.
By the lemma, we have
( *,
( )- ,
( )-+)
⏟(,
( )- ) ⏟(,
( )
( *,
This is just what we wanted to prove. ◘
( )- ,
( )-+)
( ) ( )
( )- )
( )