Introduction
Let X and Y be discrete random variables, then the conditional expectation of X given the event Y=y is a function of y over the range of Y
where is the range of X.
A problem arises when we attempt to extend this to the case where Y is a continuous random variable. In this case, the probability P(Y=y) = 0, and the Borel–Kolmogorov paradox demonstrates the ambiguity of attempting to define conditional probability along these lines.
However the above expression may be rearranged:
and although this is trivial for individual values of y (since both sides are zero), it should hold for any measurable subset B of the domain of Y that:
In fact, this is a sufficient condition to define both conditional expectation and conditional probability.
Read more about this topic: Conditional Expectation
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