Markov kernel: Difference between revisions

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Revision as of 09:43, 18 July 2023

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===[[Galton–Watson process]]===

As a less obvious example, take <math>X = \N, \mathcal A = \mathcal P(\N)</math>, and <math>(Y, \mathcal B)</math> the real numbers <math>\R</math> with the standard sigma algebra of [[Borel set]]s. Then

:<math>\kappa(B|n)=\begin{cases} \mathbf{1}_B(0) & n=0\\ \Pr(\xi_1 + \cdots + \xi_x \in B) & n \neq 0 \\ \end{cases} </math>{{Clarify|reason=what is $x$?|date=May 2022}}

withwhere <math> x </math> is the number of element at the state <math> n </math>, <math>\xi_i</math> are [[Independent and identically distributed random variables|i.i.d.]] [[random variable]]s <math>\xi_i</math> (usually with mean 0) and where <math>\mathbf{1}_B</math> is the indicator function. For the simple case of [[Bernoulli distribution|coin flips]] this models the different levels of a [[Galton board]].

== Composition of Markov Kernels and the Markov Category==