Imagine I want to randomly sample how fast a driver is going on a highway. And for this purpose, let's assume that the distribution is normal with the mean as the speed limit. Now if I take a sample, that would give me a single speed that a driver might be going. But what I'd like to do is repeatedly sample how fast one specific driver is going. My thought process is to first sample on the previously mentioned distribution to get that specific driver's average speed, u1. Then perform many samplings on the new normal distribution with mean=u1.
Please look at comments for a better example.
My feeling is that for repeated samples, each driver has their own individual mean, which the first sample gives us, and then further samples should be around that mean and not the global mean.
I feel like this is the right thing to do, but I can't express the idea well to get any hits on google. Is this the correct way to do this (and if so what's the name for this technique)?