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Assuming there are $n$ uncorrelated random variables (RVs) with Rice (Rician) distributions $R1~N(u_1,s_1) \ldots R_n~N(u_n,s_n)$, with non-zero mean and different variance, what is the mean and variance of $Z= \displaystyle \frac{R1+\ldots+Rn}{n}$?

Thank you!

Amzoti
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Min
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  • $$E[Z] = \frac{1}{n}\sum_{i=1}^n E[R_i]$$ $$\operatorname{var}(Z) \frac{1}{n^2}\sum_{i=1}^n \operatorname{var}(R_i)$$ The fact that the random variables are Rician has nothing to do with these results. The mean formula applies universally; the variance formula requires that the random variables be uncorrelated. – Dilip Sarwate Mar 06 '13 at 03:06
  • That last formula should read $\displaystyle\operatorname{var}(Z) = \frac{1}{n^2}\sum_{i=1}^n \operatorname{var}(R_i).$ – Dilip Sarwate Mar 06 '13 at 03:30
  • @DilipSarwate Thanks for your comments. From the last formula, can it be said that the uncertainty of an average with n elements improves as n grows? – Min Mar 06 '13 at 03:51
  • @DilipSarwate I mean, does the variance of Z decrease as n grows? – Min Mar 06 '13 at 04:01
  • Here is my more detailed question.

    What is the mean and variance of $Z= \displaystyle \frac{R_1+\ldots+R_n}{n}$, where $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 $\sigma_i$, $Y_i$ is a random variable with Gaussian with zero mean and standard deviation of $\sigma_i$. I believe the probability density function for $R_i$ is a Rician distribution but can't figure out its mean and variation.

     – Min Mar 06 '13 at 05:08

1 Answers1

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check out the following wikipedia link http://en.wikipedia.org/wiki/Rice_distribution and use the commented hint

Learner
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