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Conditional Expectations and Distributions
Suppose
is a discrete r.v. Now we define the conditional
distribution of another r.v.
given
(at some point where
) by
By the law of total probability,
As we discussed before, when we condition on an event, we are shrinking the sample space under consideration. So there is some normalization that takes place.
We also define
Note that this depends on the value of ; it is a function of . Let us now take the expectation with respect to :
We can think of as a discrete random variable that is a function of .
For a discrete r.v.
, the function
could be either
a discrete or a continuous r.v. Discrete:
Continuous: There exists a function such that
We can also write
If
is discrete we have
When is a continuous r.v., conditional probabilities and expectations are somewhat more complicated, because for any particular value of .
Recall that
for some function
, and
.

It can be shown that under the stated conditions, such a function
always exists.
 If is discrete then as defined earlier satisfies the property.
 is unique, in the sense that if there are two functions and both satisfying ( 2 ) then .
Once we have defined conditional expectation, we can define a
conditional c.d.f.:
Properties:
 This definition agrees with the previous one when is discrete.
 .
 is a c.d.f. as a function of because it satisfies all the properties of a c.d.f.

If
and
are jointly continuous then
has a
density for every
,
If and are jointly continuous then exists and
Also,
Analogously for continuous random variables