Deriving the second raw moment of the rice PDF

kstephe6

(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.

kstephe6

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

kstephe6

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.