Notes
Mathematical notes in probability and related areas.
Study notes Random Variables as Vectors in L²
In-progress notes formalizing random variables as vectors in the Hilbert space \(L^2(\Omega,\mathcal{F},\mathbb{P})\). They interpret expectation and conditional expectation as orthogonal projections, variance and bias-variance as Pythagorean decompositions, and Chapman-Kolmogorov as a tower-property identity.
$$ \mathcal{H}=L^2(\Omega,\mathcal{F},\mathbb{P}),\qquad \langle X,Y\rangle=\mathbb{E}[XY],\qquad \mathbb{E}[X\mid Y]=\operatorname{Proj}_{\mathcal{H}_Y}(X) $$