Exchangeability and de Finetti\u27s theorem

Končno zaporedje slučajnih spremenljivk je zamenljivo, če je porazdelitev zaporedja nespremenjena za vsako permutacijo indeksov. Neskončno zaporedje ${ X_i } _{i in mathbb{N}}$ slučajnih spremenljivk je zamenljivo, če so končna zaporedja $X_1,...,X_n$ zamenljiva za vsako naravno število $n$. Če so slučajne spremenljivke zamenljive, potem so tudi enako porazdeljene. Obratno v splošnem ne velja. Velja, ko imamo neskončno zaporedje zamenljivih slučajnih spremenljivk. Očitno pa velja v primeru, ko so enako porazdeljne slučajne spremenljivke tudi neodvisne. De Finettijev izrek pravi, da je zamenljivo neskončno zaporedje Bernoullijevih slučajnih spremenljivk ‘mešanica\u27 neodvisnih zaporedij pogojno na mero $mu$ na $[0,1]$.A finite sequence of random variables is exchangeable if the distribution of the sequence is unchanged for every permutation of the indices. Infinite sequence ${ X_i } _{i in mathbb{N}}$ of random variables is exchangeable, if the finite sequences $X_1,...,X_n$ are exchangeable for every natural number $n$. If the random variables are exchangeable, then they are identically distributed. In general the opposite does not hold. It holds if we have an infinite sequence of exchangeable random variables. It is obviously true in the case that identically distributed random variables have independent property as well. De Finetti\u27s theorem says that an exchangeable infinite sequence of Bernoulli random variables is a ‘mixture\u27 of independent sequences conditional on measure $mu$ on $[0,1]$