Assuming there's a variable I want to measure, but I have only very noisy instrument to do so. So I
Finley Mckinney
Answered question
2022-06-10
Assuming there's a variable I want to measure, but I have only very noisy instrument to do so. So I want to take multiple measurements so that I have a better chance to recover the state of the variable. Hopefully, with each measurement, my instrument can report the result as a Gaussian distribution , with the mean to be the most likely state of variable and the standard deviation suggests a rough possible region of the state. My problem now is that I don't know how to combine these multiple measurements to get a sensible answer. My guess is that it would be nice if I can get a new gaussian from these results, with the mean centered at the expectation value of the state of the variable, and a standard deviation to reflect how confident I am about the result... I tried to teach myself about gaussians, and probabilities, but I just couldn't get my head around...please can someone help me?
Answer & Explanation
livin4him777lf
Beginner2022-06-11Added 14 answers
Actually, you are measuring something of the form
where v is the deterministic value you want to measure, and is the value of a Gaussian noise at the th measurement. If the measurements are independent from each other, then simply take the arithmetic mean
which has a normal distribution, by linear combination of Gaussian variables. If your system is ergodic (broadly speaking, the system and the noise do not change behavior over time, i.e. v and the distribution of do not change over time), then the expected value and the variance of are
The random variable has a normal distribution . If you have a reference measurement where the value is known, e.g. deduced from another measurement technique, then you can estimate and deduce by how much the noise modifies the mean of . If the noise distribution changes at each measurement, for each i, then the arithmetic mean has a normal distribution . Alternatively, one can compute the weighted and centered mean
which reduces to the arithmetic mean when and for all i. The expected value and the variance of are