Inferring Poisson $\lambda$'s and then deducing their ratio

We are now faced to the inference of $r_1$ and $r_2$ from the observed numbers of counts and the observation times $T_1$ and $T_2$. For simplicity we start assuming $T_1=T_2$, so that we can focus on $\lambda_1$, $\lambda_2$ and their ratio. The extension to the general case will be straightforward, as we shall see from Sec.  on.