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cqfd
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Hi everyone, I'm new here and this is my first post in this forum. ^^
Suppose that you observe a fluorescent object whose true location is x0. Individual photons come from this object with apparent locations xi in an approximately Gaussian distribution about x0. The root-mean-square deviation σ of this distribution mostly indicates the wavelength of light, not the true size of the object. But you are interested in the object's position, not its size. To estimate the true mean x0, you decide to take finite sample {x1, ... , xN} and compute its averate <x>N
Even if the object is not moving, nevertheless each time you do the measurement, you'll find a slightly different value for <x>N. Find the root-mean-square deviation of <x>N from the true mean x0.
Gausian distribution: P(x) = 1/sqrt(2Pi)/σ*exp(-(xi-x0)^2/2σ^2)
What I've done so far is to compute <x>N = Ʃxi*Gaussian by using the definition of the mean. This is the final form I computed and I didn't find any way to simplify.
Then I assumed that because the measurements are done several times, that <x>N becomes
<x>N,j.
Then I inserted this in the definition of the RMS deviation, which got me this:
RMS = sqrt(<(<x>N,j-x0)^2>).
But this is like a dead end because, honestly, I have no idea how I could solve this analytically. Could you give me a hint or a trick on how I could solve this?
Homework Statement
Suppose that you observe a fluorescent object whose true location is x0. Individual photons come from this object with apparent locations xi in an approximately Gaussian distribution about x0. The root-mean-square deviation σ of this distribution mostly indicates the wavelength of light, not the true size of the object. But you are interested in the object's position, not its size. To estimate the true mean x0, you decide to take finite sample {x1, ... , xN} and compute its averate <x>N
Even if the object is not moving, nevertheless each time you do the measurement, you'll find a slightly different value for <x>N. Find the root-mean-square deviation of <x>N from the true mean x0.
Homework Equations
Gausian distribution: P(x) = 1/sqrt(2Pi)/σ*exp(-(xi-x0)^2/2σ^2)
The Attempt at a Solution
What I've done so far is to compute <x>N = Ʃxi*Gaussian by using the definition of the mean. This is the final form I computed and I didn't find any way to simplify.
Then I assumed that because the measurements are done several times, that <x>N becomes
<x>N,j.
Then I inserted this in the definition of the RMS deviation, which got me this:
RMS = sqrt(<(<x>N,j-x0)^2>).
But this is like a dead end because, honestly, I have no idea how I could solve this analytically. Could you give me a hint or a trick on how I could solve this?