### Understanding the Log-Normal and Log-Gamma Distributions Parameterizations

Tuesday, September 11th, 2012Naturally, when dealing with some particular probability distribution that fits to many of your data sets well, one day you will want to learn more about that distribution. What is so specific to it that it works for your data? And, moreover, how can you interpret the distribution parameters?

The good news is: for many probability distributions, the meaning of their parameters is described in the scientific literature. The classic example is the Normal distribution having two parameters: σ (scale) and μ (location). The parameterization of this distribution is pretty easy to understand: as you change the location parameter, the probability density graph moves along the x-axis, while changing the scale parameter affects how wide or narrow the graph is:

However, for quite a few of distributions, modifying the parameters and observing how the graphs change will be of little help for your undetstanding of what those parameters indicate. One of such distributions is the Lognormal model defined as “a continuous probability distribution of a random variable whose logarithm is normally distributed.” What does that mean exactly? For better understanding, compare the CDFs of the Normal and Lognormal distributions:

*Normal distribution CDF*

*Lognormal distribution CDF*

As you can see, the Normal model is “included” into the Lognormal in such a way that in the Lognormal model, ln(x) has the Normal distribution with the same parameters (σ, μ) as the original Lognormal distribution. And this is the key point to understand: the parameters of the Lognormal model are not the “pure” scale and location (pretty intuitive in the Normal model), but rather the scale and location of the included Normal distribution.

The same logic applies to the Gamma and Log-Gamma pair of distributions. The classical Gamma has two parameters: α (shape), β (scale), and the Gamma CDF is as follows:

*Gamma distribution CDF*

The shape parameter indicates the form of the Gamma PDF graph, while the scale factor affects the spread of the curve. Similarly to the Gamma model, the Log-Gamma distribution has two parameters with the same names (α, β), but its CDF has the form:

*Log-Gamma distribution CDF*

Just like with the Normal & Lognormal analogy, in the Log-Gamma model, ln(x) has the Gamma distribution with the same parameters (α, β) which cannot be treated as the “pure” Log-Gamma shape and scale, but the shape and scale of the included Gamma model.

There are dozens of different probability distributions out there, and even if you use only a couple of them on a daily basis, sometimes it can be hard to remember the meaning of all the parameters. That is why we decided to include a little feature in EasyFit that helps you keep your memory fresh: when moving the mouse pointer over a distribution parameter edit box, EasyFit displays a pop-up hint indicating the meaning of that particular parameter:

*Distribution parameter hint*

Using this feature, you can better focus on your core analysis rather than the technical details like the ones outlined in this article.