Brownian Bridge™

1. In which space does the projection interpretation of conditional expectation hold?

L¹L²C⁰Almost surely

2. What property characterizes the projection?

Martingale propertyMaximum likelihoodOrthogonality of the residualAlmost sure convergence

3. Complete the statement: $E[X|\mathcal{G}] = \arg\min_{Y \in L^2(\mathcal{G})}$ ___

E[|X-Y|]

E[(X-Y)^2]

\text{Var}(X-Y)

P(X \neq Y)

4. It is sufficient to check orthogonality only against indicator functions of G-sets.

truefalse

Quiz: Conditional Expectation as an L² Projection