# How can this CDF be decreasing? (power-law)

I am reading the first vignette of the powerRlaw R package.

It uses the moby dataset, which is an array of word frequency. Each element is a word, and the value if the frequency of that word in the Moby Dick novel.

At page 5 there is a plot of the CDF for this dataset:

There is a similar plot in the article

Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf’s law. Contemporary Physics, 46(5), 323–351. https://doi.org/10.1080/00107510500052444

at page 6:

Now Wikipedia says that CDF are increasing functions, starts at 0 and end up at 1.

How can this CDF be decreasing?

• The first figure that you have posted, y axis is incorrect. It should be called CCDF (complementary cumulative distribution function) or survival function. May 29, 2021 at 13:33
• The $y$-axis scale on the second chart is not a probability (it exceeds $1$) - in fact it is the number of distinct words which appear at least $x$ times, so goes up to $18855$ on the left hand side May 29, 2021 at 22:18

The vignette by Gillespie (2020) cites Clauset et al (2009) as another source of the figure. Reading that latter paper, it looks like they are actually plotting the complementary CDF (better known as the survival function), which is $$P(x) = 1-F(x) = \mathbb{P}(X>x)$$. This latter function is of course non-increasing rather than non-decreasing. Both papers are sloppy in their references to this function (they really shouldn't call it the CDF), but that appears to be their intention.