Can't stress this enough- in Canada (Ontario specifically) it is a serious fine for anyone caught using the term without being certified by a professional organization.
All provinces have a comparable professional organization, all of which are members of Engineers Canada. We have the notion of "self-regulated professions", where the government gives a charter to the professional organization to regulate use of their name, their members, etc. For Americans: it's not comparable to the IEEE, it's comparable to the College of Physicians and Surgeons (also a Canada/UK thing the US doesn't really have). So they're given the same power to penalize people "impersonating engineers" as the royal college can penalize people impersonating doctors.
This is a perfect example of why the term is protected by PEO. I used to think the PEO overreaching in its attempts to protect the term/designation, but I can see why they have taken issue with the term "Software Engineer" being genericized.
I applaud the effort to learn the topics of "statistics, probability, and linear algebra", but these would have been relatively fundamental courses in most software/computer/electrical engineering curricula that I've known about, and most definitely a prerequisite to calling oneself an engineer.
In some states, which regulate such things. I don't know about fines, but if a gang from the IEEE or the ASME catch you wandering around wearing their colors without being a member,....