Careful there, Godric- if you're going to quote facts at least make sure they're accurate. In POA, it actually states that "Sirius and James transformed into such large animals..." [1] and that "They could each turn into a different animal at will." [2] They can transform, at will, but it is not stated whether they chose their forms.
However, in the definition of an animagus as given on the Harry Potter Wiki, it clearly states that:
An Animagus (pl. Animagi) is a witch or wizard who can turn into an animal at will. Learning how to become an Animagus is difficult; it took James Potter, Sirius Black, and Peter Pettigrew the better part of two years to figure out how to do it. The form one takes on as an Animagus reflects their personality. Animagi are required to register with the Ministry of Magic, to ensure they do not cause too much mischief while in their animal form. [3]
[1] Harry Potter and the Prisoner of Azkaban, U.K. Edition: page 260
[2] Harry Potter and the Prisoner of Azkaban, U.K. Edition: page 259
[3] Harry Potter Wiki: http://harrypotter.wikia.com/wiki/Animagus