Lecture 4 — Self-check answers

1. The measure on (Borel) subsets of the range obtained by pushing \(P\) through \(X\) via inverse images; the distribution / law of \(X\).
2. \(P_{X}(x)=P(X^{-1}((-\infty,x]))=P(\{\omega:X(\omega)\le x\})\).
3. They generate the Borel \(\sigma\)-algebra; RVs are set up so those inverse images are events we can assign \(P\) to.
4. The unbounded rectangle of points with first coordinate \(\le x_1\) and second \(\le x_2\) (“southwest” of \((x_1,x_2)\)).
5. Nature / labelers assign the underlying measure; we estimate distributions from samples in the range of RVs.
6. Probability is a measure of sets; a density value at one point is not \(P\) of a singleton in continuous models.
7. One joint experiment vs two correlated experiments (disease process vs imaging/labeling process) — modeler’s choice.