Deriving the second raw moment of the rice PDF


by kstephe6
Tags: deriving, moment, rice
kstephe6
kstephe6 is offline
#1
Mar5-12, 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.
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kstephe6
kstephe6 is offline
#2
Mar5-12, 10:43 AM
P: 3
FYI, this is exactly the same problem as finding <x^2> (the average of the square) for P(x)
kstephe6
kstephe6 is offline
#3
Mar7-12, 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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