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Deriving the second raw moment of the rice PDF 
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#1
Mar512, 09:40 AM

P: 3

(the following is Mathematica code)
For the rice pdf... P(x)=(x/σ^2) BesselI[0, x (ν/σ^2)] Exp[(x^2 + ν^2)/(2 σ^2)] ...the Second raw moment is given by doing the integral... Integrate[(x^3/σ^2) BesselI[0, x (ν/σ^2)] Exp[(x^2 + ν^2)/(2 σ^2)], {x, 0, Infinity}] While the answer is known to be... 2 σ^2 + ν^2 ...I have no idea how to arrive at this answer by hand (ie without using Mathematica). I am currently doing a little research for grins and giggles on a very closely related integral, and knowing how to solve the one above would be of great use to me. Any help in this endeavor... Even suggestions as to where I might find the derivation... would be greatly appreciated. 


#2
Mar512, 10:43 AM

P: 3

FYI, this is exactly the same problem as finding <x^2> (the average of the square) for P(x)



#3
Mar712, 05:39 AM

P: 3

Never mind... I figured it out. The answer is very lengthy, so If anyone is curious just let me know. Otherwise, I'm not taking the time to post 3 pages of recursive formulas and integration by parts.



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