It will depend on the university. Where I studied, an introductory class to probability and statistics was mandatory, as was measure theory where probability th

It will depend on the university. Where I studied, an introductory class to probability and statistics was mandatory, as was measure theory where probability theory was at least hinted at. Other universities may have even stronger probability theory requirements, or none at all. But it's certainly not an uncommon specialisation. Note that probability theory is not necessarily the same thing as statistics. While the latter builds upon the former, statistics is more about "given the data, what's the most likely distribution ", as opposed to "given the distribution, what's the data gonna look like" (probability theory). For mathematicians, the latter seems to be more relevant, as it's a more deductive form of reasoning.