Given an event
Let be an event with . Given a discrete Random variable , the conditional expectation , given , is expressed as
If is continuous, the expectation becomes an integral instead.
Here, is called the conditional PDF and is defined as the derivative of the conditional CDF .
Given a random variable
Let and be random variables. Let be a function that takes any real number and produces . Then the conditional expectation of , given , is given by the random variable defined by .
The distinction between what is a constant and what is a random variable is important.
- remains a constant.
- is also a constant.
- is a random variable. It is defined as , which is a transformation of , a random variable (here, we don’t know that for some ).
Properties
- If and are Independent, then .
- for any function . Intuitively, we are given , so we can treat as a constant and move it out under expectation properties. In other words, and .
- .