Randomly generate clustered points given a center coordinate in 3D

I would like to be able to generate a cluster of points in 3D space that would create a majority of the points within a specified sphere radius (in this case, 4) from the starting point (in this case, 0,0,0). I'd also like it to generate the occasional outlier too.

It would be similar to this:

But I'd like to be able to tweak some of the settings. I will be generating this in C# and I'm familiar with the Random class, and have some familiarity with spheres. I'm not sure how I can piece it all together.