##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics

## J. Schweinsberg

#### UCSD

## Family size distributions for multitype Yule processes

##### Abstract:

Qian, Luscombe, and Gerstein (2001) introduced a model of gene duplication that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are no deaths and each individual gives birth to a new individual at rate one. When a new individual is born, it has the same type as its parent with probability $1 - r$ and is a new type, different from all previously observed types, with probability r. We refer to individuals with the same type as families and provide an approximation to the joint distribution of family sizes when the population size reaches $N$. We also show that if $1 << S << N^{1-r}$, then the number of families of size at least S is approximately $CNS^{-1/(1-r)}$, while if $N^{1-r} << S$ the distribution decays more rapidly than any power.

Host:

### October 7, 2004

### 10:00 AM

### AP&M 6438

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