What is the best way to combine the results of multiple uncertain measurements?
For example, let us assume that I want to measure the relation y ~ b*x. I run my experiment and I estimate the the parameter 'b', with a mean of 10 and sd = 2, normally distributed.
To increase my certainty I run the experiment again. This time I estimate 'b' with a mean of 10.5 and a sd of 1.5, normally distributed.
What is the correct way to combine these results? It seems intuitive to me that the new mean would be closer to 10.5 than 10, but perhaps I'm wrong. I suppose also that the sd of the combined parameter will be lower - I will be more confident about my measurement.
If I wanted to add the measurements of a third experiment, I'm assuming my certainty in the result would increase again. Lets assume that there is no additional information in these measurement processes. Each measurement is exactly the same and weighted equally.
How would one go about this for the normally distributed measurements? What is general process for any type of distribution? Is there an official name for this process?