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What is the mean and variance of $Z_0=\frac{\sum\nolimits_{i = 1}^{{n}}R_iZ_i}{\sum\nolimits_{i = 1}^{{n}}R_i}$, where $Z_i$ is a constant and $R_{i} =\sqrt{X_i^2+Y_i^2} $ ${(i=1,\ldots ,n)}$? $X_i$ is a random variable with Gaussian with mean of $u_i$ and standard deviation of $\sigma_i$, so $X_i\sim N(u_i,\sigma_i)$. $Y_i$ is a random variable with Gaussian with zero mean and standard deviation of $\sigma_i$, so $Y_i\sim N(0,\sigma_i)$.

I believe the probability density function for $Z_0$ is a Rician distribution but can't figure out its mean and variation. Also, I would like to show that the variance of $Z_0$ becomes narrower as $n$ grows.

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Your $R_i$ are Rician random variables and it is not immediately obvious that $Z_0$ is a Rician random variable. Could you provide some indication why you believe that $Z_0$ has a Rician distribution? –  Dilip Sarwate Apr 25 '13 at 13:31
    
I can be wrong but I am pretty sure that will not be a Rician distributed random variable. –  Caran-d'Ache Aug 5 '13 at 6:35

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