Deriving the second raw moment of the rice PDF

AI Thread Summary
The discussion focuses on deriving the second raw moment of the Rice probability density function (PDF) using an integral approach. The integral provided involves the Bessel function and exponential terms, with the known result being 2σ² + ν². The original poster expresses difficulty in solving the integral by hand and seeks guidance on the derivation process. Ultimately, they mention that they have solved it but find the answer lengthy and are not inclined to share the detailed solution. The conversation highlights the complexity of integrating functions involving Bessel functions and the Rice distribution.
kstephe6
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(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.
 
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FYI, this is exactly the same problem as finding <x^2> (the average of the square) for P(x)
 
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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