Statistics and probability

Random variables, the Gaussian, Brownian motion, Poisson, Bayes.

Probability is the language the bookshelf uses whenever a quantity is random rather than deterministic. Three places where this is unavoidable: the thermal motion of air molecules in Sound 1.3 (Brownian motion), the spike trains of auditory-nerve fibres in Hearing Ch 5 (Poisson processes), and the brain’s inferential reading of noisy sensory data in Hearing Ch 8 (Bayesian perception). All three rest on the same probabilistic vocabulary.

This chapter is that vocabulary. Five lessons covering the named distributions you’ll meet, the Central Limit Theorem that makes the Gaussian inescapable, the random-walk picture that becomes Brownian motion and the diffusion equation, the Poisson process that governs random arrivals, and the Bayesian / signal-detection machinery for inferring underlying state from noisy observations.

• 11.1 Random variables and distributions — PMF, PDF, CDF, expected value, variance, the six named distributions you’ll see (uniform, Bernoulli, binomial, Gaussian, exponential, Poisson).
• 11.2 The Gaussian and the central limit theorem — why the bell curve is everywhere; the CLT and its demonstration on a sum of i.i.d. samples; the multivariate Gaussian.
• 11.3 Random walks and Brownian motion — i.i.d. random steps; mean-square displacement growing as √N; continuum limit to Brownian motion and the diffusion equation; the Einstein relation linking diffusion to viscosity.
• 11.4 Poisson processes — random arrivals at constant rate; the exponential inter-arrival distribution; the Poisson count distribution; superposition and thinning.
• 11.5 Bayesian inference and signal detection — Bayes’ rule; prior × likelihood = posterior; conjugate Gaussian updates; signal detection theory; ROC curves and $d'$.

If you haven’t done probability in a while, this chapter is the audience it was written for. Each lesson reintroduces its central idea before any algebra.