When I was in the 3rd grade of my middle school, I enjoyed my town bookstore as a standing library. There a series of six math-story books by Zhang Yuan-Nan impressed me a lot. I cited a case from one in my PPT when I taught the normal distribution -- the normal pdf can be derived from simple symmetry & independence conditions.
Today I can even google out an illegal pdf of its new edition to verify the case (2005, pp. 89). Actually I have bought the new edition series (now 3 books) and lent them to students. Those conditions are as instinctive as--
1. For white noise errors on 2-D, the independence means pdf at is the product of 1-D pdf, that is, .
2. The symmetry means pdf at is just a function of , nothing to do with direction. That is, .
For middle school students, the book stated a gap here to arrive at the final result , which is .
I think non-math graduate students with interests can close the gap by themselves with following small hints.
That is, .
Now to prove . With continuousness, it gets .