My answer to a question on Quora:

The answer is very simple: Statistics and Mathematics have different sources of funding. From time to time, mathematical disciplines get separated from mathematics because their end users (industry, the military, the government) wish to pay only for the end product, and do not wish to subsidise its mathematics roots. This has happened with computer science, cryptography, fluid dynamics, etc. — and statistics is just one of many examples. There is no any intrinsic reason for all these disciplines to be separate from mathematics.

Below is a fragment from a book written by my university lecturer in mathematical statistics on the bases of lecture notes taken by his PhD students (they were attending for that purpose our undergraduate lectures). This is the very first chapter, the first definition: The notion of a sample.

As you can see from this example, mathematical statistics can be the purest of pure mathematics.