I am trying to compute the mean of the i-th order statistics for \(\displaystyle n\) Rayleigh random variables as follows:
. . . . .\(\displaystyle \large{ \displaystyle \int_0^{\infty}\, A \, x \, F^{i-1} (1\,-\,F)^{n-i} \, f \, dx }\)
where
. . . . .\(\displaystyle \large{ A\, =\, i\, \binom{n}{i} \, \mbox{ and }\, F\,=\,1\,-\,e^{\frac{-x^2}{2\sigma^2}} }\)
is the CDF of the Rayleigh RV and
. . . . .\(\displaystyle \large{ f\,=\,\frac{x}{\sigma^2}e^{\frac{-x^2}{2\sigma^2}} }\)
is its pdf.
I start by substituting \(\displaystyle p\,=\,1\,-\,F\), changing the integral limits to between \(\displaystyle 0\) and \(\displaystyle 1\) and then substituting the values of
. . . . .\(\displaystyle \large{ \displaystyle x\, =\, \left(-2\,\sigma^2\, \ln(P)\right)^{\frac{1}{2}}\, =\, \sigma\,\sqrt{\strut 2\,}\left(\ln\left(\frac{1}{p}\right)\right)^{1/2} }\)
The integral becomes:
. . . . .\(\displaystyle \large{ \displaystyle A\, \sqrt{\strut 2\,}\, \sigma\,\int_0^1 \, \left(\ln\left(\frac{1}{p}\right)\right)^{1/2} \,(1\,-\,p)^{i-1}\, P^{n-i}\, dp }\)
Then, by using integration by parts:
. . . . .\(\displaystyle \large{ \displaystyle -A \,\sqrt{\strut 2\,} \, \sigma \, \Gamma\left(\frac{3}{2}\right)\, \bigg[ (n\,-\,i) \,\beta (n-i,\,i) \,-\, (i\,-\,1) \,\beta(n\,-\,i\,+\,1,\, i\,=\,1)\bigg] }\)
where \(\displaystyle \Gamma\) is the Gamma function and \(\displaystyle \beta\) is the Beta function.
In the integration by parts, I use
. . . . .\(\displaystyle \large{ \displaystyle u\,=\, (1\,-\,p)^{i-1}\,P^{n-i} \, \mbox{ and }\, dv\,=\, \left(\ln\left(\frac{1}{p}\right)\right)^{1/2} \, dp }\)
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?
. . . . .\(\displaystyle \large{ \displaystyle \int_0^{\infty}\, A \, x \, F^{i-1} (1\,-\,F)^{n-i} \, f \, dx }\)
where
. . . . .\(\displaystyle \large{ A\, =\, i\, \binom{n}{i} \, \mbox{ and }\, F\,=\,1\,-\,e^{\frac{-x^2}{2\sigma^2}} }\)
is the CDF of the Rayleigh RV and
. . . . .\(\displaystyle \large{ f\,=\,\frac{x}{\sigma^2}e^{\frac{-x^2}{2\sigma^2}} }\)
is its pdf.
I start by substituting \(\displaystyle p\,=\,1\,-\,F\), changing the integral limits to between \(\displaystyle 0\) and \(\displaystyle 1\) and then substituting the values of
. . . . .\(\displaystyle \large{ \displaystyle x\, =\, \left(-2\,\sigma^2\, \ln(P)\right)^{\frac{1}{2}}\, =\, \sigma\,\sqrt{\strut 2\,}\left(\ln\left(\frac{1}{p}\right)\right)^{1/2} }\)
The integral becomes:
. . . . .\(\displaystyle \large{ \displaystyle A\, \sqrt{\strut 2\,}\, \sigma\,\int_0^1 \, \left(\ln\left(\frac{1}{p}\right)\right)^{1/2} \,(1\,-\,p)^{i-1}\, P^{n-i}\, dp }\)
Then, by using integration by parts:
. . . . .\(\displaystyle \large{ \displaystyle -A \,\sqrt{\strut 2\,} \, \sigma \, \Gamma\left(\frac{3}{2}\right)\, \bigg[ (n\,-\,i) \,\beta (n-i,\,i) \,-\, (i\,-\,1) \,\beta(n\,-\,i\,+\,1,\, i\,=\,1)\bigg] }\)
where \(\displaystyle \Gamma\) is the Gamma function and \(\displaystyle \beta\) is the Beta function.
In the integration by parts, I use
. . . . .\(\displaystyle \large{ \displaystyle u\,=\, (1\,-\,p)^{i-1}\,P^{n-i} \, \mbox{ and }\, dv\,=\, \left(\ln\left(\frac{1}{p}\right)\right)^{1/2} \, dp }\)
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 a moderator: