Deriving the second raw moment of the rice PDF

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SUMMARY

The discussion focuses on deriving the second raw moment of the Rice probability density function (PDF) using the integral of the form Integrate[(x^3/σ^2) BesselI[0, x (ν/σ^2)] Exp[-(x^2 + ν^2)/(2 σ^2)], {x, 0, Infinity}]. The known result for this integral is 2 σ^2 + ν^2. The original poster sought assistance in deriving this result by hand, as they were exploring related integrals. Ultimately, they resolved the problem independently but noted that the derivation involves lengthy recursive formulas and integration by parts.

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  • Understanding of Rice probability density function (PDF)
  • Familiarity with Bessel functions, specifically BesselI[0]
  • Knowledge of integration techniques, including integration by parts
  • Proficiency in Mathematica for computational verification
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  • Study the derivation of moments for the Rice distribution
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Mathematicians, statisticians, and researchers working with probability distributions, particularly those interested in the Rice distribution and its moments.

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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