Notes In progress
Random Variables as Vectors: Geometry of L², Projections, Variance, and Bias
Summary
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.
$$ \mathcal{H}=L^2(\Omega,\mathcal{F},\mathbb{P}),\qquad \langle X,Y\rangle=\mathbb{E}[XY],\qquad \mathbb{E}[X\mid Y]=\operatorname{Proj}_{\mathcal{H}_Y}(X) $$
Context
The notes form a conceptual bridge from linear algebra to probability: once random variables are treated as L² vectors, expectation, conditional expectation, variance identities, and Markov transition formulas become projection statements.
Main contributions
- Defines random variables as measurable functions and builds the \(L^2\) inner-product geometry using \(\mathbb{E}[XY]\).
- Shows expectation and conditional expectation as orthogonal projections onto constants and information subspaces.
- Interprets variance, bias-variance, total variance, and prediction error as Pythagorean decompositions.
- Derives Chapman-Kolmogorov from indicators, the Markov property, and the tower property of conditional expectation.