Pure Birth Process

Definition

A Pure Birth Process is a birth and death process where ${\mu }_n=0$ for all $n$ (no deaths occur).

The process only moves from state $n$ to state $n+1$ with birth rate ${\lambda }_n$. The transition rates are:

$\mathrm{Pr}\{X(t+h)=n+1\mid X(t)=n\}={\lambda }_{n}h+o(h)$ $\mathrm{Pr}\{X(t+h)=n\mid X(t)=n\}=1-{\lambda }_{n}h+o(h)$

Interpretation

A pure birth process models a population or system that only grows over time—members can be added but never removed. The Poisson process is the simplest case where ${\lambda }_n=\lambda$ (constant birth rate).

Special Cases

• Poisson Process: ${\lambda }_n=\lambda$ for all $n$ (constant rate)
• Yule Process: ${\lambda }_n=n\lambda$ (linear rate, each member gives birth independently)

Related

• Poisson Process
• Yule Process
• Birth and Death Process
• Continuous-Time Markov Chain

Exercises

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Kuis 2 2024 No. 2. Pure birth process dari $X(0)=0$ dengan ${\lambda }_0=1$, ${\lambda }_1=3$, ${\lambda }_2=2$, ${\lambda }_3=5$.

(a) Tentukan $\mathbb{E}[W_3]$ dan $\text{Var}(W_3)$.

Jawaban: $W_3=T_0+T_1+T_2$ dengan $T_i\sim \text{Exp}({\lambda }_i)$. $\mathbb{E}[W_3]=\frac{1}{1}+\frac{1}{3}+\frac{1}{2}=\frac{11}{6}$. $\text{Var}(W_3)=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{2^2}=\frac{49}{36}$.

(b) Tentukan mean dari $W_1+W_2+W_3$.

Jawaban: $\mathbb{E}[W_1+W_2+W_3]=(\frac{1}{1})+(\frac{1}{1}+\frac{1}{3})+(\frac{1}{1}+\frac{1}{3}+\frac{1}{2})=1+\frac{4}{3}+\frac{11}{6}=\frac{6+8+11}{6}=\frac{25}{6}$.