Statistics and probability

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

Probability is the language for quantities that are random rather than deterministic — the thermal motion of air molecules (Sound 1.3), the spike trains of auditory-nerve fibres (Hearing Ch 5), the noisy sensory data the brain reads to infer the world (Hearing Ch 8). The five lessons below cover the named distributions, 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 $d'$.

Each lesson reintroduces its central idea before any algebra.