The polarization of random variables is an important concept in statistics. New theorems regarding polarized Bernoulli random variables are presented on the following topics: (1) The polarization of the maximal correlation parameter; (2) The graphical model of polarized variables; (3) The Wyner common information between variables. In addition, the application of polar code ensembles for broadcast channels is discussed. To complement our recent theory, experimental results are presented for Goela-Abbe-Gastpar broadcast codes achieving rates on the capacity boundary of several types of broadcast channels for which no other low-complexity codes are known.