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PROB · 1

Probability spaces and random variables

2 concepts · 2 questions

Kolmogorov's axioms, events as measurable sets, random variables as measurable functions, distributions and distribution functions, and expectation as a Lebesgue integral.

Definition 1.1Probability space and random variables

A probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is a measure space with $\mathbb{P}(\Omega) = 1$ (Kolmogorov's axioms). Events are sets in $\mathcal{F}$ . A random variable is a measurable $X : \Omega\to\mathbb{R}$ . Its distribution is the measure $\mu_X(B) = \mathbb{P}(X\in B)$ on $\mathbb{R}$ , determined by the distribution function $F_X(x) = \mathbb{P}(X\le x)$ (non-decreasing, right-continuous, limits $0$ and $1$ ).

Probability = measure theory with total mass 1. Events are measurable sets; random variables are measurable functions.

Why measure theory: it handles discrete, continuous and mixed variables, and infinite sequences of coin tosses, in one framework.

Definition 1.2Expectation

$\mathbb{E}[X] = \int_\Omega X\,\mathrm{d}\mathbb{P}$ (defined if $\mathbb{E}|X| < \infty$ ). Change of variables: $\mathbb{E}[g(X)] = \int_\mathbb{R}g\,\mathrm{d}\mu_X$ , which is $\sum g(x)p(x)$ for discrete $X$ and $\int g(x)f(x)\,\mathrm{d}x$ when $X$ has density $f$ . Linearity, monotonicity, and MCT/DCT hold. For $X\ge0$ : $\mathbb{E}[X] = \int_0^\infty\mathbb{P}(X > t)\,\mathrm{d}t$ . $\operatorname{Var}X = \mathbb{E}[(X - \mathbb{E}X)^2] = \mathbb{E}X^2 - (\mathbb{E}X)^2$ .

Expectation = Lebesgue integral against $\mathbb{P}$ . Discrete sums and density integrals are special cases.

Tail formula: $\mathbb{E}X = \int_0^\infty\mathbb{P}(X > t)\,\mathrm{d}t$ for $X\ge0$ .

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