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Expectation  ::  Properties  ::  Pairs  ::  Independence  ::  Two R.V.S.  ::  Functions  ::  Inequalities  ::  Conditional  ::  General

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 .

1. It can be shown that under the stated conditions, such a function always exists.

2. If is discrete then as defined earlier satisfies the property.
3. is unique, in the sense that if there are two functions and both satisfying ( 2 ) then .
When a condition is true with probability 1, we say that it is true ''almost surely,'' or ''a.s.''

Once we have defined conditional expectation, we can define a conditional c.d.f.:

Properties:
1. This definition agrees with the previous one when is discrete.
2. .
3. is a c.d.f. as a function of because it satisfies all the properties of a c.d.f.
4. If and are jointly continuous then has a density for every ,

There is another interpretation:

If and are jointly continuous then exists and

Also,

Analogously for continuous random variables

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). More on Random Variables. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec2_8.html. This work is licensed under a Creative Commons License