Q:

Proper way to deal with several sources of variability

What is the proper way of combining various sources of variability? For example, if we know that under the normal conditions, a machine produces parts with normally distributed diameter with some $\mu$ and $\sigma$, and that measuring this diameter has its own normally distributed "noise" with $\sigma_{meas}$. We than sample $n$ parts and measure each exactly once. 
What would be the mean and standard deviation of the measurements under the assumption that the machinery works fine?
I am solving a homework assignment. I have stripped down the assignment to what I think is its pivotal point, from which I will be able to work my way out. 

A:

Generally, when you look at the final measurements results you always look at the sums of the particular diameter and a measurement error. Both are random variables with given means and standard deviations.
Now, generally for two random variables
$$
Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y).
$$
If X and Y are independent, then the covariance is zero and it becomes easier. Does this help you?

