I measured something N times using different measurement techniques. Each measurement technique i has a known variance σ i 2 . So every measurement is x i = x ^ + ϵ i where x ^ is the true value and ϵ i is pulled from a normal distribution with mean 0 and variance σ i 2 .Ok, I have all these measurements. Now I want to know the variance of my measurements taken together, taking into account the fact that I know the variance of all the measurement techniques. What's the best way to do this?

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I measured something   N times using different measurement techniques. Each measurement technique   i has a known variance       σ    i    2  . So every measurement is       x    i    =            x      ^        +      ϵ    i   where             x      ^       is the true value and       ϵ    i   is pulled from a normal distribution with mean 0 and variance       σ    i    2  .Ok, I have all these measurements. Now I want to know the variance of my measurements taken together, taking into account the fact that I know the variance of all the measurement techniques. What&#039;s the best way to do this?

Alexia Hart · Accepted Answer

You can just treat them as sum of normally distributed variables.In short, if   X  ∼  N  (  x  ,      σ    x    2    ) and   Y  ∼  N  (  y  ,      σ    y    2    ) and   X  ,  Y are independent, then  X  +  Y  ∼  N  (  x  +  y  ,      σ    x    2    +      σ    y    2    )In your case you have the measurements       X    i    ∼  N  (  x  ,      σ    i    2    ), but you don&#039;t know the actual value   x (which is why you&#039;re doing the measurements.) Now In order to find   x, you probably want to take the average of all of them, that is             x      ^        =      1    n        ∑          i      =      1        n        X    i  If we now treat             x      ^       as a random variable, we can determine its variance by using the formula above, and we get:  n            x      ^        =      ∑    i        X    i    ∼  N  (  n  x  ,      ∑    i        σ    x    2    )Now as a last step you have to &quot;solve&quot; for             x      ^       which is not difficult. (Use that   V  a  r  (  a  X  )  =      a    2    V  a  r  (  X  ), and   E  [  a  X  ]  =  a  E  [  x  ])

I measured something N times using different measurement techniques. Each measurement techniqu

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