Yale University’s new president will be Maurie D. McInnis, currently the president of Stony Brook University, a New York state public university where she is known for raising the school’s profile, donations and prestige.

When she takes over from Yale’s outgoing president, Peter Salovey, in July, Dr. McInnis, who earned masters and doctoral degrees from Yale in the 1990s, will become the university’s first permanent female president.

Her selection followed a search that began last…