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There are 4 measures for the characterization of the shape of a probability distribution: expectation (1st order raw moment), variance (2nd order central moment), skewness (expression in 3rd and 2nd order central moments) and kurtosis (expression in 4th and 2nd order central moments).

I intend to find a probability distribution whose mean, variance, skewness and kurtosis can be freely manipulated: like with 'sliders'.

Anybody can see the effect of say variance alone on the appearance of the distribution, without changing the kurtosis of it.

I went ahead with some initial workouts. They are attached below.

Workout for the 4 measures

Note that the summation or the integral version of the expression of the raw moments applies as per whether we are working on the discrete or the continuous case of the problem, respectively.

Any leads after this?

  • N.B.: I know that there's an impossible region of (skewness, kurtosis) tuples. I also know that probability distributions aren't just random shapes adjusted to a certain set of values of those 4 characters: they need to be meaningful and applicable to practical scenarios; perhaps most often, bell shaped in some way or the other. I also admit that my writing may contain numerous mistakes: please feel free but not obliged to mark out to the tiniest of them - from preposition errors to incorrect terminology.
BeBlunt
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    By "independent" I presume you don't mean "statistically independent" - especially since it seems you're talking about the population moments. But then I am unclear quite what the word is intended to imply. What specific sense of 'independent' is this? – Glen_b Aug 07 '22 at 07:38
  • I mean if I were to create a program that could simulate the change of the curve wrt the values provided. I could toggle the "variance" slider to 4 from 3, changing the shape of the curve, however without changing the value reflected on the "kurtosis" slider. So I currently wish to find an explicit equation for a curve like that, whose characters maybe changed without one impacting another. – BeBlunt Aug 07 '22 at 08:59
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    There are several families of distributions that you might do this with; the first such are the Pearson distributions. If you specify the first four moments (assuming you specify a possible pair of the third and fourth moments), you will specify a particular member of the Pearson family, which have 6 main subtypes. The diagram here shows the square of skewness and kurtosis for the Pearson family, indicating the main subtypes and the names of some distributions which are of that type – Glen_b Aug 07 '22 at 09:56
  • (There's a couple of additional things that were added to the plot which are not specifically Pearson family, however). Oh, it might be 7 main subtypes actually, depending on what you count as 'main' – Glen_b Aug 07 '22 at 10:01
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    There have also been multiple families of distributions suggested with the aim of being able to specify population skewness and kurtosis parameters, for a number of purposes (e.g. with the aim of random number generation) – Glen_b Aug 07 '22 at 10:07
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    Please bear in mind there are infinitely many measures of a distribution's shape and they are not all determined by the four mentioned at the outset of this question. This opens up the possibilities if your underlying objective is to have a flexible way to specify a shape. – whuber Aug 07 '22 at 13:13
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    One approach to what it is that it seems you are trying to do is given here: https://stats.stackexchange.com/a/483215/102879 – BigBendRegion Aug 07 '22 at 15:25
  • If you want to move the "lever" and see what happens to the distributions, you have to constrain yourself to a particular family of distributions, wherein the distribution can be uniquely determined by the first four moments. Two examples are given here: https://math.stackexchange.com/a/2523606/472987 . In the first family, the distributions become flatter as you increase the kurtosis lever, interestingly enough. – BigBendRegion Aug 07 '22 at 18:48

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