Conditional expectation generalized to continuous random variables
The conditional expectation definition needs to be generalized, to be applicable to continuous random variables. The generalization is accompanied with an example and later will be applied to expected shortfall.
Generalizing conditional expectation definition
Suppose can take values with probabilities and can take values with probabilities
Denote the joint probabilities . The definition from this post gives
where is a fixed value of . The drawback of this definition is its dependence on indexation of values . Our purpose is to show how from this definition one can obtain a definition that does not use indexation and can therefore be applied to continuous random variables. Denote
Then the sum can be expanded by including zero terms:
(the sum in the middle includes all points in the sample space). Using (1) and (3) we can rewrite (2) as
Replacing here the conditioning on by conditioning on a general set whose probability is not zero we obtain the definition of conditional expectation:
Example. If is standard normal and is its distribution function, then for any number one has
where is the density.
Proof. From the expression of the density
Applying this equation and (4) we get