Conditional Density Function
whenever .
Definition 4.25 (Conditional Density Function)
Let and be two random variables with joint density and marginal densities and respectively.
The conditional density function of given is defined as
Law of total probability for densities
If and have joint density and marginal densities and , then for every ,
Conditional expectation via the conditional density
Conditioning on defines a conditional expectation by averaging against the conditional density: where
As in the discrete case, is a random variable that is a function of .
Related
Stated in
- Definition 4.25 (Conditional Density Function)§4.6.4 Conditional Density Functions