Lecture 3 — Self-check answers

1. \((\Omega,\mathcal{F},P)\); analogy \((\mathbb{R},\text{nice subsets},\text{length})\).
2. \(\Omega\) may be non-numeric/abstract; algorithms need numbers/vectors.
3. A (measurable) function \(X:\Omega\to\mathbb{R}\) or \(X:\Omega\to\mathbb{R}^{d}\).
4. It is a deterministic function, not a “random” object nor an ordinary algebra “variable.”
5. Abstract: disease/visit/imaging process as outcomes in \(\Omega\); concrete: stored image vector in \(\mathbb{R}^{d}\).
6. The image is a point in the range of a random variable defined on an underlying sample space.
7. How \(P\) pushes forward to a measure/distribution on (subsets of) \(\mathbb{R}^{d}\).