I don't understand what 'independent of position' means. Can you give some examples?
Hi Alan, by independent of positon I mean that the statistical moments, mean, variance, etc of the probability density function describing the mass density (apologies for the double use of density, but it should be clear) don't depend on the position on the shell. I'm not assuming that they are point masses inside a sphere. The standard result uses symmetry to calculate the acceleration due to gravity, starting with a diagram similar to trackstar's post, except the point where "m" is, is inside the shell, eg Figure 2 here https://www.math.ksu.edu/~dbski/writings/shell.pdf
. If the mass is not evenly distributed then I can't use symmetry to give us a 1-D integral for the acceleration.
In reply to Paul, if the mass density is constant on the shell, then the gravitational field is zero in the shell. If the mass density wasn't constant, but could be characterised by a probability density function, and I'll assume all moments exists for the point of the argument, then it seems plausible that there's a fluctuation of the gravitational field inside the shell, but the expected value would be zero. is there a dipole or quadrupole term due to the uneven distribution of mass on the shell.
It seems like a plausible problem, but the discussions, lecture notes and references I can find assume that the density of mass on the shell is constant.