Yule-Simon distribution

1. Sablon:Matematika ?

The Yule-Simon distribution models phenomena with power-law behavior, such as word frequencies or city sizes. Its probability mass function is:

$$P(k)=\alpha \text{Beta}(k,\alpha +1)$$

with $\alpha >0$ as the shape parameter. It has a heavy tail, meaning a few events are very common while many are rare. It’s used in areas like linguistics (Zipf’s law), ecology, and economics.

Key stats:

• Mean: $\frac{\alpha }{\alpha -1}$ (if $\alpha >1$)
• Variance: $\frac{{\alpha }^2}{(\alpha -1{)}^2(\alpha -2)}$ (if $\alpha >2$)