# How can I obtain the function described by given set of central moments?

I want to investigate how my function P behaves with different probability functions rho as input variables. This means

Output = P[rho]


I have to do this in a scientific and systematic way. One way could be:

1. Observing that each propability function can be described by central moments

2. Start with the inspection of how the the output changes with the first central moment (expected value) while keeping all other moments constant.

3. Study how the Output changes with the second central moment (=variance), letting all other moments constant.

4. .....

I want to find out which moments are relevant and which are not.

1. Do you have alternative suggestions or experience with this approach?

2. How can I create a function which has given central moments (for example, mu = 1, sigma = 1, gamma = 0, kurtosity = 2)? Is there a way to do it with Mathematica?

I have not studied mathematics, but as far as I can understand, there should be a solution for every set of central moments as long the HankelMatrix is positive-semi-definit (Hausdorff's moment problem).

But how do I get this solution?

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(1) Not all probability distributions are determined by central moments; moreover, some sets of moments give rise to more than one probability distribution. (2) You are not free to fix a finite set of central moments and vary the rest: there are relationships among them all that must be satisfied. This suggests you consider alternative characterizations of distributions. But which ones would be appropriate? As to that, you have provided us no clues. Perhaps you could tell us what your problem is rather than stating it in such abstraction that we have little chance of helping you? –  whuber Mar 4 '13 at 17:34
Thanks for your answer. The theoretic problem and the algorithm are complicated, but I will try to give more hints. The optimal debt level d of a company is destermined by the tax rate s, insolvency costs ic and the return probability function rho of a company (More return --> more debt, less return -> less debt) (This is called the Trade-Off-Theory'). I generate for a given probability function a 3D-Grid (optimal debt level d in dependence od tax rate s and insolvency costs ic). So I can explain the dependence of the optimal debt level of s and ic. –  Peter Mar 4 '13 at 23:20
The problem: I have to find out and describe (in a scientific way) if and how the output 3D-Grid changes, if I do not use the NormalDistribution function. My program is general, so I can use every function as Input. In a paper I can not just choose 5 random functions and plot the 3D grid for them. I need to argue how these grid changes with different functions (or is the result relatively independent of the propability distributions). My guess would be that only the first 2 Moments are critical, the other do not change much. That would be a easy-understandable result. –  Peter Mar 4 '13 at 23:22