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Let $A$ and $B$ be independent and normally distributed variables where $A \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $B \sim \mathcal{N}(\mu_2,\sigma_2^2)$.

What will be the density of $A^2 + B^2 -2AB\cos(\phi)$, where $\phi$ is a constant?

I read the answer in Finding the distribution of $5X_{1}^2+2X_{1}X_{2}+X_{2}^2$ , applying the same trick will result in the sum of two scaled non-central and dependent chi-squares.

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Anas Alhashimi
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  • I can't quite understand the question. Are you asking about the pdf of the sampling distribution of the Euclidean distance (a scalar) between two correlated variables (realized in some sample size `n`) with correlation $\phi$? – ttnphns Feb 22 '17 at 13:24
  • Actually I have a sensor that measures the distances and angles of two arbitrary placed land marks $L_A$ and $L_B$. I would like to find an unbiased estimator for the distance between the two land marks $d_{AB}$. So I have the distance measurements are normally distributed and the angle difference $\phi$ assumed noiseless for simplicity. – Anas Alhashimi Feb 22 '17 at 16:00
  • The same "trick" applies to your situation, doesn't it? All you have to do is change the numbers. – whuber Feb 22 '17 at 23:48

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