(a) A colony of bacteria evolves as follows. Let $X$ be a random variable with values in the positive integers. Each bacterium splits into $X$ copies of itself after an exponentially distributed time of parameter $\lambda>0$. Each of the $X$ daughters then splits in the same way but independently of everything else. This process keeps going forever. Let $Z_{t}$ denote the number of bacteria at time $t$. Specify the $Q$-matrix of the Markov chain $Z=\left(Z_{t}, t \geqslant 0\right)$. [It will be helpful to introduce $p_{n}=\mathbb{P}(X=n)$, and you may assume for simplicity that $\left.p_{0}=p_{1}=0 .\right]$

(b) Using the Kolmogorov forward equation, or otherwise, show that if $u(t)=$ $\mathbb{E}\left(Z_{t} \mid Z_{0}=1\right)$, then $u^{\prime}(t)=\alpha u(t)$ for some $\alpha$ to be explicitly determined in terms of $X$. Assuming that $\mathbb{E}(X)<\infty$, deduce the value of $u(t)$ for all $t \geqslant 0$, and show that $Z$ does not explode. [You may differentiate series term by term and exchange the order of summation without justification.]

(c) We now assume that $X=2$ with probability 1 . Fix $0<q<1$ and let $\phi(t)=\mathbb{E}\left(q^{Z_{t}} \mid Z_{0}=1\right)$. Show that $\phi$ satisfies

$$\phi(t)=q e^{-\lambda t}+\int_{0}^{t} \lambda e^{-\lambda s} \phi(t-s)^{2} d s$$

By making the change of variables $u=t-s$, show that $d \phi / d t=\lambda \phi(\phi-1)$. Deduce that for all $n \geqslant 1, \mathbb{P}\left(Z_{t}=n \mid Z_{0}=1\right)=\beta^{n-1}(1-\beta)$ where $\beta=1-e^{-\lambda t}$.