Module: Conditional Expectation

Learning Outcomes

Conditional Expectation

1. There are three concepts in this module:
   • Conditional pmf/pdfs

   • Taking expectations with respect to (conditional) pmf/pdf on \(X\) conditioned on an event \(A\), $${\mathcal E} [ X

     A]$$

   • The special random variable: $${\mathcal E} [X

     Y]\(, a function of\)Y\(, and not a nubmer like\){\mathcal E} [ X

     A]$$

2. If we want to estimate a rv \(X\) (that we do not get to see), but make an observation \(Y\), then \({\mathcal E}[X|Y]\) is the best estimate of \(X\) (in the mean square sense)

3. Covariance, correlation, conditional variance

4. Law of iterated expectations \({\mathcal E}\bigl[{\mathcal E}[X|Y]\bigr] = {\mathcal E} X\)

5. $${\mathcal E}[X	Y]\(is uncorrelated with the error\)X- {\mathcal
   E}[X	Y]\(made in estimating\)X\(from\)Y$$

Readings

Experiential Learning