4.5 Conditional Expectation Function (CEF),

4.5.1 Part I

Think back to remember the definition of the expectation of \(Y\):

\[ E[Y] = \int_{-\infty}^\infty y \cdot f_{Y}(y) dy \]

This week, in the async reading and lectures we added a new concept, the conditional expectation of \(Y\) given \(X=x \in \text{Supp}[X]\):

\[ E[Y|X=x] = \int_{-\infty}^\infty y \cdot f_{Y|X}(y|x) dy \]

4.5.2 Part II

  1. What desirable properties of a predictor does the expectation possess (note, this is thinking back by a week)? What makes these properties desirable?
  2. Turning to the content from this week, how, if at all, does the conditional expectation improve on these desirable properties?

4.5.3 Part III

  • Compare and contrast \(E[Y]\) and \(E[Y|X]\). For example, when you look at how these operators are “shaped”, how are their components similar or different?3

  • What is \(E[Y|X]\) a function of? What are “input” variables to this function?

  • What, if anything, is \(E[E[Y|X]]\) a function of?

  1. Note, when we say “shaped” here, we’re referring to the deeper concept of a statistical functional. A statistical functional is a function of a function that maps to a real number. So, if \(T\) is the functional that we’re thinking of, \(\mathcal{F}\) is a family of functions that it might operate on, and \(\mathbb{R}\) is the set of real numbers, a statistical functional is just \(T: \mathcal{F} \rightarrow \mathbb{R}\). The Expectation statistical functional, \(E[X]\) always has the form \(\int x f_{X}(x)dx\).)↩︎