Suppose the number of sales, N, in a year on a sales portfolio has a poisson distribution with parameter $\lambda$. Sales are either large with probability $p$ or small with probability $1-p$, independently from each other.
Suppose we observer $r$ large sales. Show that the conditional distribution of $N - r \mid r$ is poisson.
I've managed to complete the question from first principles:
Let $R$ be the number of large sales, we have $ R \mid N$ is binomially distributed with parameters $N$ and $p$.
If we consider $P(N - r = s \mid R = r)$ and apply Bayes conditional probability formula we eventually get the result.
I am wondering if there is perhaps a simplier way to do it? For instance, could we perhaps just directly use the the result that, $$ \text{ Posterior PDF } \propto \text{Prior PDF } \times \text{ Likelihood}$$ that is, $$f(\theta \mid x) \propto f(\theta) \times L(x \mid \theta)$$
Can this general formula be applied in this case as a shortcut?