Developing normal pdf from symmetry & independence

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, .

So, .

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.

Denote .
We have
,
or
  .
Denote .
That is, .

Now to prove . With continuousness, it gets .

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