LogNormal

Some kind of probability distribution that is related to the normal distribution.

Basic Mathematical Properties of the Lognormal Distribution: http://www.inf.ethz.ch/personal/gut/lognormal/maths/maths.html

Generating method

A Lognormal distribution can be generated by an iterative multiplicative process. Consider the equation

X_t = F_t * X_t-1

which relates the new value of a variable to the old value multiplied by a random variable F. If F_t are independent variables with idential distributions (for all t), then the distribution of X will be Lognormal[1].

Check out TheStructuralCauseOfFileSizeDistributions for a little more on this generating method in the context of file sizes.

The Relationship between LogNormal and PowerLaw distributions

Power Laws and Lognormal distributions are quite closely related. According to [1], "as long as there is a bounded minimum that acts as a lower reflective barrier to the multiplicative [generating] model, it will yield a power law instead of a lognormal distribution."

It seems that power laws and lognormal distributions are nearly identical at high values. However lognormal distributions include less smaller values. This manifests as a downwards curve below the straight line of a power law on the right of a ranked log-log scaled graph.

Graphs

Figure 1: ranking graph of a lognormal distribution, plotted on a log-log scale. The left side of the curve is approximately a straight line, just like a power law. The right hand side curves down due to the lack of smaller values.

Figure 2: The very left side of the lognormal density function. A power law density function would continue upwards to the left, whereas this curve drops to 0.

Figure 3: The lognormal density function again, this time with the x axis using a log scale. The resulting curve is a Gaussian/normal curve. So the distribution of the log of the random variable is a normal distribution, hence the name lognormal.

A power law curve would keep rising up to the left and not fall to 0.

References

[1] A Brief History of Generative Models for Power Law and Lognormal Distributions: http://citeseer.nj.nec.com/553345.html