is a function defined on the members of the sigma field. Here is a link from Wikipedia explaining sigma fields with examples. The book by Grinstead and Snell is available online and it should be consulted for reference on probability review.
This is a rather important costruct. Probability is a function defined on the members of the sigma field. Here is a link from Wikipedia explaining sigma fields with examples.
Here is a Maple file 
which finds the power set of a sample space.
Here is a simple example of Maple inverting a p.g.f.
This Maple file finds the p.d.f. of the sum of two r.v.'s in two different ways.
In this worksheet, Maple inverts a Laplace transform.
Here we have a pair of r.v.'s X and Y whose joint density is given. The problem is to find the expected value of X given Y. The Maple solution is given here. (I don't like this example. I will give you another one on Sep. 23, 2013.)
Here is a better, and more practical, explanation of the conditional densities using an example of throwing darts.
We will discuss an example where the thief (good guy) is imprisoned in a dungeon with three doors which he can't see. One of the doors leads to freedom. If he chooses a wrong door he will return to the dungeon. What is the expected time until freedom? This problem needs conditioning arguments.
A 1924 movie starring Douglas Fairbanks has the same title as our problem.