Yule Process

Definition

A Yule Process is a pure birth process with linear birth rate:

${\lambda }_n=n\lambda ,n\ge 0$

where each member of the population independently gives birth at exponential rate $\lambda$.

The total birth rate when there are $n$ individuals is $n\lambda$, since each of the $n$ individuals acts independently.

Interpretation

Named after G. Yule, who used it in his mathematical theory of evolution. Each existing member has the same chance of producing a new member per unit time. The more members there are, the faster the population grows — exponential growth in expectation.

Properties

• Transition rates: $q_{n,n+1}=n\lambda$, $q_{n,n}=-n\lambda$
• Transition probabilities: $P_{n,n+1}=1$
• The sojourn time in state $n$ is $\text{Exp}(n\lambda )$ with mean $\mathbb{E}[T_n]=1/(n\lambda )$

Related

• Pure Birth Process
• Birth and Death Process
• Poisson Process

Exercises

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Yule Process — expected population. Misalkan proses Yule dengan ${\lambda }_n=n\lambda$ dimulai dari $X(0)=1$. Tentukan $\mathbb{E}[X(t)]$.

Jawaban: $\mathbb{E}[X(t)]=e^{\lambda t}$. Dari persamaan forward: $\frac{d}{dt}E[X(t)]=\lambda E[X(t)]$, dengan $E[X(0)]=1$. Solusi: $E[X(t)]=e^{\lambda t}$.

Yule Process — sojourn time. Hitung $\mathbb{E}[T_n]$, waktu yang dihabiskan di state $n$.

Jawaban: $\mathbb{E}[T_n]=\frac{1}{n\lambda }$. Karena laju transisi keluar dari state $n$ adalah $n\lambda$.