# PearsonDistribution[a1,a0,b2,b1,b0]: What are the formulas for a1 and a0?

I primarily use MATLAB. I use the function pearspdf(mean, std, skew, kurtosis) from this add-on to generate numbers from a particular distribution in the Pearson System. However, I am working on a problem where I have formulas for the mean, std, skew, and kurtosis and I would now like to do more than sample numbers.

Therefore, I have turned to Mathematica. Mathematica is different, in the sense that its function departs not from the moments, but from the coefficients a1, a0, b2, b1, and b0. Doing a bit of research, I discovered that b2, b1, and b0 are indeed functions of the moments, so I am making headway. But I am stumped on what a1 and a0 are, and I am hoping someone here can fill me in? I didn't find the explanation at Wolfram.com satisfactory.

UPDATE: I missed a crucial part of the documentation at Wolfram.com. Thank you for pointing this out, J.M.. See the image below:

UPDATE: So, I have tried to implement the solution provided above, my first step being to find the values for Beta1 and Beta2, implied by:

Next, I computed the values of the first four central moments from formulas I am using. These gave the results:

mu = 0

sigma = 284.60

lambda = 0

kappa = 2.8

Substituting these values, along with the computed values for Beta1 and Beta2 thus provides:

Next, I double-checked the MATLAB solution with the provided central moments, to verify the Pearson Distribution existed for the stated central moments, which it does.

However, plugging the Pearson coefficients (a1, a0, b2, b1, b0) into PearsonDistribution, and plotting, provided an error warning, that the coefficient combination is not valid.

I suspect there is something wrong with bo, but the formula is taken from the solution in the first image.

UPDATE: To make matters even more confusing, I now find that Rose and Smith in their book "Mathematical Statistics with Mathematica" (2002) have the following definitions of Beta1 and Beta2 (p.149):

Can anyone help me out here?

• By pearspdf(), you mean this add-on, right? I spent some time searching the docs before realizing it wasn't built-in or in a standard toolbox. It also seems you missed this part of the docs. – J. M. will be back soon Sep 15 '17 at 10:01
• @J.M., Yes that is right. I don't think the MATLAB function is part of a toolbox. Regarding what you think I missed, you are right, I did. I have posted an image to show the solution. Am I right in interpreting this solution as general for Pearson Systems? In other words, a0, a1, b0, b1, and b2 are defined as stated? – user120911 Sep 15 '17 at 19:32
• Did you look at the lines after that showing how to verify that the appropriate central moments are reproduced? See this earlier thread as well. – J. M. will be back soon Sep 15 '17 at 22:17
• Yes, thank you, J.M. – user120911 Sep 19 '17 at 9:37
• OP writes: "I have formulas for the mean, std, skew, and kurtosis". What are the values you have for the mean, the variance $\mu_2$, the 3rd central moment $\mu_3$ and 4th central moment $\mu_4$? – wolfies Sep 21 '17 at 14:48