What is probability space?

Example 22.2

The standard laws and their transforms of note: Bernoulli $\mathcal B(p)$, binomial $\mathcal B(n, p)$, geometric, Poisson $\mathcal P(\lambda)$ (discrete: Year 2’s tables remain valid); uniform on $\intcc01$ (Lebesgue measure itself); exponential $\mathcal E(\lambda)$ (density $\lambda\eu^{-\lambda x}\mathbf 1_{x>0}$); the Gaussian $\mathcal N(m, \sigma^2)$ with density $\frac1{\sigma\sqrt{2\pi}}\exp\bigl(-\frac{(x - m)^2}{2\sigma^2}\bigr)$ — a probability density by Problem 10.1, with mean $m$ and variance $\sigma^2$ (Gaussian moments, Exercise 11.10).

Example 22.14 (What the strong law buys)

(a) Frequencies: for i.i.d. coin flips, the observed frequency of heads converges a.s. to $p$ — the empirical justification of probability itself. (b) Monte Carlo: for $g \in L^1(\intcc01)$ and $(U_n)$ i.i.d. uniform (Theorem 22.6), $\frac1n\sum_{k\leq n}g(U_k) \to \int_0^1g$ a.s.: integrals by sampling, in any dimension, at the dimension-independent rate $\sim n^{-1/2}$ made precise in Chapter 23. (c) Normal numbers: almost every real number has, in its binary expansion, asymptotic frequency $\frac12$ of ones (apply the strong law to the digit variables of Theorem 22.6) — Borel’s theorem, a statement about everyday numbers proved by measure: Problem 22.1 completes it in all bases.