Download Some remarks about conditional expectation.

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Remarks about conditional expectation. ⋆
For a given event A with P (A) > 0 and a random variable X, the conditional expectation,
also called conditional mean, of X given A can be def ined as
E(X|A) = E(X1{A} )/P (A)
where
1 if A happens
0 otherwise.
Notice that 1{A} , called indicator function, is a random variable. This is a very commonly
used mathematical trick to transform an event into a random variable.
Example. Let X be the total number of dots of two dice and A be the event that both
dots are even numbers. Then A consists of (2, 2), (4, 4), (6, 6) three outcomes out of 36
outcomes. Therefore P (A) = 3/36 = 1/12.

4 for outcome (2,2) with probability 1/36


8 for outcome (4,4) with probability 1/36
X1{A} =

 12 for outcome (6,6) with probability 1/36
0 else with probability 1-3/36=11/12
1{A} =
n
So E(X1{A} ) = (4 + 8 + 12)/36 = 2/3, as a result,
E(X|A) = E(X1{A} )/P (A) = (2/3)/(1/12) = 8.
The above definition is in my view easier and more helpful in understanding the following
derivation in Example 3.6:
E(T1 1{X2 =0} |X1 = 0) = E(T1 |X2 = 0, X1 = 0)P (X2 = 0|X1 = 0)
because the right hand side, by definition, is
=
E(T1 1{X2 =0,X1 =0} )
E(T1 1{X2 =0,X1 =0} ) P (X2 = 0, X1 = 0)
×
=
P (X2 = 0, X1 = 0)
P (X1 = 0)
P (X1 = 0)
E(T1 1{X2 =0} 1{X1 =0} )
= E(T1 1{X2 =0} |X1 = 0)
P (X1 = 0)
again by the definition of conditional expectation. We note that 1A∩B = 1A 1B for any
two events A and B.
An alternative but equivalent way of defining conditional expectation is to use conditional
distribution. Define the conditional distribution as F (x) = P (X ≤ x|A). The conditional
expectation of X given A is then
Z ∞
E(X|A) =
xdF (x).
−∞
Conditional expectation is a very important concept is probability, which is further
generalized in advanced probability.