The theorem says that the distribution of an interchangeable sequence of Bernoulli-distributed random variables can be viewed as an integral over conditionally independent Bernoulli-distributed random variables.
Formulation of the sentence
Let be an infinite sequence of interchangeable Bernoulli-distributed random variables with parameters and density . Then there is a probability distribution with a distribution function , so that for each and every realization :
,
where the number of "successful" Bernoulli attempts is.
Consideration as weighting
In other words, we can say that there exists a random variable on with distribution function , so that the given independent conditionally are, that is
and it also applies to everyone
.
Furthermore, de Finetti's law of large numbers applies