From outcomes to distributions

Let (\Omega,\mathcal F,\mathbb{P}) be a probability space. A random variable is a measurable function from outcomes to values. For the stipulated four-form pronoun space, a case variable can be defined by

C(\omega) \equiv \begin{cases} \mathrm{nom},&\omega\in\{\textit{he},\textit{they}\},\\ \mathrm{acc},&\omega\in\{\textit{him},\textit{them}\}. \end{cases}

Its value space and sigma-algebra are

\mathcal X_C\equiv\{\mathrm{nom},\mathrm{acc}\}, \qquad \mathcal G_C\equiv2^{\mathcal X_C}.

The distribution of C assigns each B\in\mathcal G_C the probability

\mathbb{P}(C\in B) \equiv \mathbb{P}\!\left(C^{-1}(B)\right).

As a function of B, this assignment is a probability measure on (\mathcal X_C,\mathcal G_C). We retain \mathbb{P} for the probability measure on (\Omega,\mathcal F) and introduce no second measure symbol. Expressions such as \mathbb{P}(C=c) abbreviate the corresponding preimage events. Lowercase p, q, and r denote PMFs; lowercase f, g, and h denote PDFs; uppercase F and G denote CDFs.

Random variables and one-variable distributions

We first explain why random variables are needed and then give the measurable-function definition. The next pages distinguish discrete random variables from absolutely continuous random variables and define their probability mass functions, probability density functions, and cumulative distribution functions.

Expectations and moments

The expected value is an integral with respect to a probability distribution when that integral exists. The notes then develop expectations of functions, linearity of expectation, and failure of finite expectations. Variance, standard deviation, and central moments summarize specified powers of centered values.

Distribution families

The discrete families are categorical, Bernoulli, binomial, hypergeometric, geometric, negative binomial, and Poisson. Their supports and probability mass functions differ because they represent different observation processes.

The continuous families are uniform, beta, normal, chi-squared, and Student’s t. Each page states the support, parameterization, and density or defining construction.

Joint distributions

A joint distribution preserves value pairings from the same outcome. Marginalization sums or integrates out one coordinate. Conditional independence states that one conditional distribution is invariant to an additional conditioning variable. Conditional expectation, covariance, and correlation summarize features of conditional or joint distributions.

The chapter dependency graph records the prerequisite relations among these definitions. Begin with why random variables are needed.