Theory of Intelligence
MATH FOR AI•Probability•Common Distributions•3.4 Change of Variables

Introduction

Finding the distribution of a transformed random variable: the distribution function technique, the change-of-variables formula, and inverse transform sampling.

Introduction

If XX has a known distribution and we compute Y=g(X)Y = g(X), what is the distribution of YY? This question is everywhere: standardising features, log-transforming prices, generating random numbers, and normalizing flows that turn simple noise into realistic data. Densities don't transform like values — they must be corrected by how much gg stretches space.

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