Deriving the second raw moment of the rice PDF

• kstephe6
In summary, the conversation discusses the Second raw moment for the rice pdf, which can be found by integrating a specific function. The answer is known to be 2 σ^2 + ν^2, but the speaker is looking for a way to solve it without using Mathematica. They have since figured it out, but the derivation is lengthy and they are not willing to post it unless someone is curious.
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.

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.

1. What is the rice PDF?

The rice PDF (Probability Density Function) is a mathematical function used to describe the probability distribution of a random variable. It is commonly used in statistics and probability theory.

2. What is the second raw moment of the rice PDF?

The second raw moment of the rice PDF is a measure of the spread or variability of the data. It is calculated by taking the integral of the square of the random variable over its entire range of possible values.

3. How do you derive the second raw moment of the rice PDF?

The second raw moment of the rice PDF can be derived by using the formula for moments, which involves taking the integral of the random variable raised to the desired moment (in this case, the second moment) over its entire range of possible values.

4. What is the significance of deriving the second raw moment of the rice PDF?

The second raw moment of the rice PDF is an important statistic that helps to describe the shape and spread of the probability distribution. It is used in various statistical calculations and can provide valuable insights into the data being analyzed.

5. Can the second raw moment of the rice PDF be used to make predictions?

While the second raw moment of the rice PDF is an important statistic, it is not typically used on its own to make predictions. It is often used in conjunction with other statistical measures to make informed predictions and decisions.

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