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June 22nd, 2016, 03:55 PM  #1 
Newbie Joined: Jun 2016 From: Jordan Posts: 1 Thanks: 0  computing the mean of the ith order statistics for $n$ Rayleigh random variables
I am trying to compute the mean of the ith order statistics for $n$ Rayleigh random variables as follows: $\int_0^\infty A \, x \, F^{i1} (1F)^{ni} \, f \, dx$ where $A = i \binom{n}{i}$ and $F=1e^{\frac{x^2}{2\sigma^2}}$ is the CDF of the Rayleigh RV and $f=\frac{x}{\sigma^2}e^{\frac{x^2}{2\sigma^2}}$ is its pdf. I start by substituting $p=1F$, changing the integral limits to between $0$ and $1$ and then substituting the values of $x=(2\sigma^2 \ln{ P})^{\frac{1}{2}}$=$\sigma\sqrt{2}(\ln{\frac{1}{p} })^{1/2}$. The integral becomes: $$A \sqrt{2} \sigma\int_0^1 (\ln{\frac{1}{p}})^{1/2} (1p)^{i1} P^{ni} dp$$ Then, by using integration by parts: $$ A \sqrt{2} \, \sigma \, \Gamma(3/2) \bigg[ (ni) \beta (ni,i)  (i1) \beta(ni+1, i=1)\bigg]$$ where $\Gamma$ is the Gamma function and $\beta$ is the Beta function. In the integration by parts, I use $u= (1p)^{i1} P^{ni}$ and $dv=(\ln{\frac{1}{p}})^{1/2} \, dp$ The problem is that I have run a simulation for these values and the results are different from the theoretical results I derived. Could you please help me in finding any mistakes I may have made? Last edited by Saud; June 22nd, 2016 at 03:58 PM. 

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$n$, computing, integrals, ith, order, order_statistics, random, rayleigh, statistics, variables 
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