ConditionalExpectation Summary
Intuition
E X Y a random variable
thatrepresents the provisional average
E X Y
y the value of the RV E x x
block particianof r
when Y Ay
y
E X ly is constant on each block
Ay
E x ly is 6 Y measurable
E X Y is the best 6 Y measurable RV
that estimates X
Definition of conditional expectation
Let Cr 4 IP be a probability space
X E MF be a random variable
If G F sub sigma algebra of
E 4
is a
then RV Z
any
is a verson of E xi
g if
1 measurable
, g
2 Z satisfies the partialaverage property
Sa Z dip Sa X dip for A
every Eg
These versions are unique IP almostsurely
Define as E X
g
2 Conditional expectation if X
Y
E ME
If Cr 4 IP be a probability space
B X Y E MF be a random variable
then E x 14 E XI 6 Y
Intuition
E X Y a random variable
thatrepresents the provisional average
E X Y
y the value of the RV E x x
block particianof r
when Y Ay
y
E X ly is constant on each block
Ay
E x ly is 6 Y measurable
E X Y is the best 6 Y measurable RV
that estimates X
Definition of conditional expectation
Let Cr 4 IP be a probability space
X E MF be a random variable
If G F sub sigma algebra of
E 4
is a
then RV Z
any
is a verson of E xi
g if
1 measurable
, g
2 Z satisfies the partialaverage property
Sa Z dip Sa X dip for A
every Eg
These versions are unique IP almostsurely
Define as E X
g
2 Conditional expectation if X
Y
E ME
If Cr 4 IP be a probability space
B X Y E MF be a random variable
then E x 14 E XI 6 Y
