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What would the probability density function be for a graph with input variables: mean, standard deviation, skewness, and kurtosis?

For example, if the inputs were confined only to mean and standard deviation, the formula would be:

$${\displaystyle f(x,\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}$$

(the pdf formula for a normal distribution)

I looked on https://en.wikipedia.org/wiki/Pearson_distribution and found this:

$${\displaystyle p(x)={\frac {|{\frac {\Gamma(m+{\frac {\nu}{2}}i)}{\Gamma(m)}}|^{2}}{\alpha B(m-{\frac {1}{2}},{\frac {1}{2}})}}*[1+({\frac {x-\lambda}{\alpha}})^{2}]^{-m}*e^{-\nu *arctan({\frac {x-\lambda}{\alpha}})}}$$

It seems like it could be what I'm looking for, but I am unsure as to what all the symbols mean. If someone could explain, that would be great.

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  • $\begingroup$ Pearson distributions are different from a normal distribution, if not entirely unrelated. For a normal distribution, the skewness and kurtosis are implied by the pdf. See Table, wherein skewness $=0$, excess kurtosis $=0$, and BTW kurtosis $=3$. $\endgroup$
    – Carl
    Mar 14 '20 at 1:53
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    $\begingroup$ Can you explain the sense in which specifying only the mean and the variance leads you to the normal distribution ("the formula would be...") and not some other distribution with the same mean and variance? It may be fairly important to answering your question. $\endgroup$
    – Glen_b
    Mar 14 '20 at 5:48
  • $\begingroup$ @Glen_b-ReinstateMonica I used the normal distribution as an example as opposed to some other distribution because it is the best estimate for a given distribution without knowledge of higher moments. I precisely am looking for a more general distribution formula that allows for higher moment specifications (specifically skewness and kurtosis) in addition to mean and standard deviation. This general distribution would also assume moments after kurtosis to be what they would be in a normal distribution as the best estimate for the graph of the distribution. $\endgroup$ Mar 15 '20 at 4:34
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For a distribution with parameters for the mean, sigma, skewness, and kurtosis, I can only think of the Stable (Levy) distribution with its 4 parameters (mean=$\mu$, s.d.=$c$, skewness=$\beta$, and kurtosis=$\alpha$). The stable distribution can represent normal, skew-normal, logistic, Rayleigh, Cauchy, etc.

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There are several families of distributions that can account for a variety of skewness and kurtosis levels. A good summary is

On Families of Distributions with Shape Parameters

These include:

  1. Skew-Symmetric distributions
  2. Two piece distributions
  3. Sinh-arcsinh distributions.
  4. Alpha-stable family
  5. Many other families of distributions obtained by transforming the pdf/cdf or using parametric changes of variables.
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