Pure birth process with $\lambda_n =n\lambda$

Question

In pure birth process we assume that $\lambda_n=n\lambda$

1. Write down system of equations that describe that process if $P(X(0)=1)=1$
2. Solve equations that describe $P_1, P_2, P_3$ and guess formula for $P_n$
3. Using induction check if formula for $P_n$ fulfills the equations
4. What is the formula for $P_n$ if $P(X(0)=i)=1$ for (different, any) $i \in N$

This is what i managed to do:

First i wrote down the equations:

$P_1(0)=1$

$P_n(0)=0$

$P'_1(t)=-\lambda_1P_1(t)=-\lambda P_1(t)$

$P'_n(t)=-\lambda_n P_n(t)+\lambda_{n-1}P_{n-1}(t)=-n\lambda P_n(t)+(n-1)P_{n-1}(t)$

Then I solved them for $P_1, P_2$ and $P_3$:

$P_1(t)=e^{-\lambda_1 t}=e^{-\lambda t}$

$P_2(t)= \int_{0}^{t} e^{-\lambda_2 (t-s)}\lambda_1 P_1(s) ds= e^{-2\lambda t}(e^{\lambda t} -1)=e^{-\lambda t}(1-e^{-\lambda t})$

$P_3(t)= \int_{0}^{t} e^{-\lambda_3 (t-s)}\lambda_2 P_2(s) ds= e^{-\lambda t}-e^{-3\lambda t}-2e^{-2\lambda t} +2e^{-3\lambda t}=e^{-\lambda t}-2e^{-2\lambda t}+e^{-3 \lambda t}=e^{-\lambda t}(1-2e^{-\lambda t}+e^{-2\lambda t})=e^{-\lambda t}(1-e^{-\lambda t})^2$

After reading the comments I have simplified my results and then it is easy to see that the formula for $P_n$ would be:

$P_n= e^{-\lambda t} (1-e^{-\lambda t})^{n-1}$

And in the case where $P(X(0)=i)=1$ the formula for $P_n(t)$ would be:

$P_{n}(t)={n-1\choose n-i} e^{-\lambda ti} ( 1-e^{-\lambda t})^{n-1}$

The last formula I have found in Feller's book "An Introduction to Probability Theory and Its Applications" (that was the hint from my professor).

What is left is to proof that the formula for $P_n(t)$ is true

Answer 1

Assuming that $P_{1,n}(t) = e^{-\lambda t}(1-e^{-\lambda t})^{n-1}$ (which we have already shown for the case $n=1$, we have $$ P_{1,n+1}(t) = -(n+1)\lambda P_{1,n+1}(t) +e^{-\lambda t}(1-e^{-\lambda t})^{n-1}. $$ Solving this differential equation yields $P_{1,n+1}(t)=e^{-\lambda t}(1-e^{-\lambda t})^n$, thus completing the proof by induction.

How the expression for $P_{i,n}(t)$ was derived is indeed found in the text by Feller that you referenced; was there anything unclear about the exposition?