 Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist

By Kyle Siegrist.

Online text: www.randomservices.org/random/prob/Events.html

The linked section discusses events and random variables. Its terminology includes a measurable structure with the sample space; this wiki names the underlying set and its sigma-algebra separately.

Additional section: Expected Value, 2 Additional Properties.

Cited passages: Right Distribution Function; A General Definition.

Additional section: Conditional Probability.

Cited passages: Probability Spaces, §4 Conditional Probability, Definitions and Interpretations, The Basic Definition; Basic Theory, The Multiplication Rule; Conditioning and Bayes’ Theorem, countable partition and weighted-average interpretation. Whole-textbook authorship is credited to Kyle Siegrist; no separate section byline or publication or revision date is established. The University of Alabama in Huntsville is an affiliation and project supporter, distinct from authorship; Random Services hosts the text.

Cited section: Expected Value, §6 Generating Functions, Basic Theory, The Probability Generating Function: definition, power-series expansion and absolute convergence on [-1,1]; Inversion formula f(k)=P^(k)(0)/k! and its termwise differentiation proof; Factorial moments and variance under the source’s stronger radius > 1 hypothesis; Independent-sum identity and expected-product factorization; The Moment Generating Function: M(t)=G(exp(t)) and finite-domain requirement.

Cited section: Definition: sequence of independent identically distributed jumps independent of the Poisson process; Properties: conditional mean/variance proof, generating-function conditioning proof, and PGF version; Compound Poisson Distributions: random count independent of the whole sequence; fixed-count-parameter definition.