Assuming there's a variable I want to measure, but I have only very noisy instrument to do so. So I

Finley Mckinney

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

livin4him777lf

Beginner2022-06-11Added 14 answers

Actually, you are measuring something of the form
X i = v + b i ,
where v is the deterministic value you want to measure, and b i is the value of a Gaussian noise at the ith measurement. If the measurements are independent from each other, then simply take the arithmetic mean
X ¯ n = 1 n i = 1 n X i ,
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 b i N ( μ , σ 2 ) do not change over time), then the expected value and the variance of X ¯ n are
E ( X ¯ n ) = 1 n i = 1 n E ( X i ) = v + μ , V ( X ¯ n ) = 1 n 2 i = 1 n V ( X i ) = σ 2 n .
The random variable X ¯ n has a normal distribution N ( v + μ , σ 2 / n ). If you have a reference measurement where the value v is known, e.g. deduced from another measurement technique, then you can estimate μ and deduce by how much the noise modifies the mean of X ¯ n .
If the noise distribution changes at each measurement, b i N ( μ i , σ i 2 ) for each i, then the arithmetic mean X ¯ n has a normal distribution N ( v + μ ¯ n , σ 2 ¯ n / n ). Alternatively, one can compute the weighted and centered mean
X ~ n = i = 1 n w i ( X i μ i ) with the weights w i = σ i 1 j = 1 n σ j 1 ,
which reduces to the arithmetic mean X ¯ n when μ i = 0 and σ i = σ for all i. The expected value and the variance of X ~ n are
E ( X ~ n ) = v i = 1 n w i = v , V ( X ~ n ) = i = 1 n w i 2 σ i 2 = n ( j = 1 n σ j 1 ) 2 .

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