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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.

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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?

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