Consider a population process in which the probability of transition from a state with $n$ individuals to a state with $n+1$ individuals in the interval $(t, t+\Delta t)$ is $\lambda n \Delta t$ for small $\Delta t$.

(i) Write down the master equation for the probability, $P_{n}(t)$, of the state $n$ at time $t$ for $n \geqslant 1$

(ii) Assuming an initial distribution

$$P_{n}(0)= \begin{cases}1, & \text { if } n=1 \\ 0, & \text { if } n>1\end{cases}$$

show that

$$P_{n}(t)=\exp (-\lambda t)(1-\exp (-\lambda t))^{n-1}$$

(iii) Hence, determine the mean of $n$ for $t>0$.