Skewness Kurtosis Plot for different distribution I am looking for a way to plot a skewness kurtosis plot of a range of distributions: Pearson systems, log normal, gamma , generalized gamma, Pareto
1,2,3,4 etc.
I've only found some of these distribution plots. Is it even possible to plot a skewness-kurtosis plot for all theoretical distributions? How to do it for those I need?
 A: Pearson plot
Let: 
$$\beta _1=\frac{\mu _3^2}{\mu _2^3} \quad \text{and} \quad \beta _2=\frac{\mu _4}{\mu _2^2}$$
where $\sqrt{\beta_1}$ is often used as a measure of skewness, and $\beta_2$ is often used as a measure of kurtosis. 
The Pearson plot diagram characterises distributions that are members of the Pearson family in terms of skewness ($x$-axis) and kurtosis ($y$-axis):



*

*The point at (0,3) denotes the Normal distribution.

*A Gamma distribution defines the green Type III line. The Exponential distribution is just a point on the Type III line. 

*An Inverse Gamma distribution defines the blue Type V line. 

*A Power Function [Beta(a,1)] defines the Type I(J) line. Type I nests the Beta distribution.

*Type VII (symmetrical: when $\beta_1 = 0$) nests Student's t distribution etc
Adding other distributions to the Pearson diagram
The OP asks how to add other distributions (that may not even be members of the Pearson family) onto the Pearson plot diagram. This is something done as an exercise in Chapter 5 of our Springer text: Rose and Smith, Mathematical Statistics with Mathematica. A free copy of the chapter can be downloaded here: 
http://www.mathstatica.com/book/bookcontents.html
To illustrate, suppose we want to add a skew-Normal distribution onto the Pearson plot diagram. Let $X \sim \text{skewNormal}(\lambda)$ with pdf $f(x)$:

The mean is:

... while the second, third and fourth central moments are:

Then, $\beta_1$ and $\beta_2$ are given by:

Since $\beta_1$ and $\beta_2$ are both determined by parameter $\lambda$, it follows that $\beta_1$ and $\beta_2$ are related. Eliminating parameter $\lambda$ looks tricky, so we shall use numerical methods instead. For instance, we can plot $\beta_1$ and $\beta_2$ parametrically, as a function of $\lambda$, as $\lambda$ increases from 0 to say 300:

Where will this line be located on a Pearson diagram? To see the answer exactly, we need to superimpose plot P1 onto a standard Pearson diagram. Since a Pearson plot has its vertical axis inverted, we still need to invert the vertical axis of plot P1. This can be done by converting all points {x,y} in P1 into {x,9-y}, and then showing P1 and the Pearson plot together. Doing so yields:

In summary, the black line in the diagram depicts the possible values of ($\beta_1, \beta_2$) that a $\text{skew-Normal}(\lambda)$ distribution can exhibit. We start out at (0,3) i.e. the Normal distribution when $\lambda=0$, and then move along the black line as $\lambda$ increases towards infinity.
The same method can conceptually be adopted for any distribution: in some cases, the distribution will be captured by a single point; in others, such as the case here, as a line; and in others, something more general.
Notes


*

*Expect and PearsonPlot are functions from the mathStatica add-on to Mathematica; Erf denotes the error function, and ParametricPlot is a Mathematica function.

*In the limit, as $\lambda$ increases towards infinity, at the upper extremum, $\beta_1$ and $\beta_2$ take the following values, here expressed numerically: {0.990566, 3.86918} 
