Steve Witham <sw@tiac.net> wrote:

"Keith F. Lynch" <kfl@KeithLynch.net> wrote:

...

Do you mean that the values of the three coordinates are independently normally distributed, or that the distance from the origin is? It's not obvious to me that these would give the same distribution.

Allan Wechsler <acwacw@gmail.com> answered this on 3/10, but I didn't find his answer searching for "do you mean," and also there are some extra quirks in the following response:

Ignoring constants, the normal distribution is like 1/exp(x^2).

Independent distributions for the three axes combine by multiplying:

1 / (exp(x^2) exp(y^2) exp(z^2) )= 1/exp(x^2 + y^2 + z^2)

Say r is the distance from the origin, r^2 = x^2 + y^2 + z^2.

So the distribution is like 1/exp(r^2). Except for constants. And I think the converse has to be true too.

Thanks. That's the rare explanation that not only leaves me with no doubt, but that makes me kick myself for not having come up with it myself.