Change of Variables: Same Probability, Different Area

Chapter 13 · Mathematics for AI and Machine Learning · Advanced · 1:46

An interactive live film: a small square under the 2D Gaussian and its image under a stretching map stand side by side in 3D; the patch's area grows by det J while its density drops by the same factor, so the two columns hold the same probability.

Narration

On the left, X is a standard Gaussian in the plane, a round bell. On the right is Y equals g of X, for a map g that stretches the plane more and more toward one side, and bends it.

Take a small square on the left. The probability that X lands in it is the height of the bell times the area of the square: the volume of the amber column.

The rest of the narration is in the book's Companion.