Any two individuals in a population can mate with each other producing one offspring in an exponential time with parameter $\lambda$. At time $t=0$ there are just 2 individuals ($X_0=2$).
Find the birth rate of this continuous-time stochastic process?
I am stuck finding the birth rate as it is state dependent. As two individuals are required to reproduce, the rate will change from $\frac{n}{2}\lambda$ when the population is even to $\frac{n-1}{2}\lambda$ when the population is odd. Is this ok to put as an answer or is there one birth rate for the population regardless of the state? Or am I completely wrong?
The description is not very clear. I would guess that what they are trying to convey is: when there are $n$ individuals in the population, there are $\binom{n}{2}=n(n-1)/2$ pairs of individuals. If each possible pair produces offspring at rate $\lambda$, then the birth rate with $n$ individuals is $\binom{n}{2}\lambda$.
Your interpretation of $\lfloor\frac{n}2\rfloor\lambda$ is not unreasonable if you imagine the population organised into separate pairs. But the description does say "any two individuals".