Why start with a point as the generator for random numbers when you can have a network or here at least a 3D object.
So I came up with the idea after running a simple python script plotting random small point objects. I choose the np.random.randn(). Just the normal random generator function.
From this I got the normal looking scatter plot of points.
Then I choose the particle generator in Blender and a simple sphere as the generator object.
The noise or the particle scatter plot did not look the same. So since I did not get it right by changing the settings. I just scaled the x axis of the object and then the noise looked similar.
From this I then suggest that generating random data using scaled objects are important as the normal distribution.
The normal distribution object I could make looked like a thick needle. An elongated ellipsoid.
I started with a cube and made into a sphere using the multi-resolution modifier.