The Yule-Simon distribution, named after Udny Yule and Herbert Simon, is a discrete probability with pmf

The semicolon in *f*(*k*; ρ) suggests that we think of *f* as a function of *k*, with a fixed parameter ρ. The way the distribution shows the connection to the beta function, but for our purposes it will be helpful to expand this function using

and so

Ignore the first part of the last line, ρ Γ(ρ + 1), because it doesn’t involve *k*. It helps to ignore proportionality constants in probability densities when they’re not necessary. What’s left is the (ρ + 1) falling power of *k* + ρ.

For large values of *k*, the falling power term is asymptotically equal to *k*^{ρ+1}. To see this, let *k* = 1000 and ρ = 3. Then we’re saying that the ratio of

1003 × 1002 × 1001 × 1000

to

1000 × 1000 × 1000 × 1000

is approximately 1, and the ratio converges 1 as *k* increases.

This says that the Yule-Simon distribution is a power law in the tails, just like the Zipf distribution and the zeta distribution. Details of the comparison between these three distributions are given here.