# How to draw a sample from Johnson's distribution with given skewness and kurtosis?

I came across Johnson's system of distributions. Here, I found the following figure:

The author says:

There is a unique Johnson distribution corresponding to each feasible combination of $\beta_1$ and $\beta_2$[, i.e. skewness and kurtosis].

My question is as follows. How can I draw a sample from a distribution that corresponds to prespecified values of $\beta_1$ and $\beta_2$? Is there any python or R package that can do that?

• If you're fitting to sample moments you can get odd results (not least -- but not only -- because the sample moments are quite misleading when the skewness and kurtosis are high). How are you choosing the skewness and kurtosis? (There's SuppDists::JohnsonFit and JohnsonDistribution::FitJohnsonDistribution, both should be on CRAN but the first at least is orphaned). See also stats.stackexchange.com/questions/60156/… – Glen_b Jun 2 '17 at 2:11
• Thanks for directing me to those packages. I wanted to simulate samples from distributions of various values for skewness and kurtosis. I chose the ranges [0, 10] and [-5, 5] with the step of 0.5 and planned to simulate a few samples from each feasible combination. – Milos Jun 2 '17 at 3:32
• Ah, if you have population moments you should be fine. Generating samples form Johnson distributions once you know the parameters is easy since they're transformed normals. – Glen_b Jun 2 '17 at 3:46
• Given pre-specified values of $\beta_1$ and $\beta_2$, it does not follow that you have to generate samples using the Johnson family --- you could alternatively also generate samples using the Pearson family, defined over the same space in $\beta_1$ and $\beta_2$. Depending on where $(\beta_1, \beta_2)$ is located on the chart, using the Pearson family may lead to simpler functional forms .. and the data that comes out at the end should still produce the same point in the diagram (given a large enough sample). – wolfies Jun 2 '17 at 7:32
• @wolfies I came across Johnson family when I was looking for the way to generate samples from distributions with specified $\beta_1$ and $\beta_2$, so I asked about it in the first place, but I will certainly take a look at Person family. Thanks for the info. :) – Milos Jun 2 '17 at 19:21