Problem Setting: Knowing that $N_1\sim\operatorname{Poisson}(\mu v)$ and $N_2\sim \operatorname{Poisson}(\tau v)$, where $\mu$, $\tau$, and $v$ > 0 are all unknown. We set $\theta = \frac{\mu}{\mu + \tau}$ to be the target parameter, and $\lambda = [\mu+\tau, v]$ to be the nuisance parameters. We want to find the conditional maximum likelihood estimatation of $\theta$ by considering the the distribution of $N_1$ given $N_1 + N_2$.
Solving Attempt:
First, we find the conditional distribution of $N_1$ given $N_1 + N_2$ which is
$$
P(N_1 = x_1|T=x_1 + x_2, N_2 = x_2) = \frac{P(x_1,x_2 = t - x_1)}{p(t)} = {t \choose x_1}(\frac{\mu}{\mu + \tau})^{x_1}(1-\frac{\mu}{\mu + \tau})^{t-x_1} \sim Bio(x_1+x_2,\frac{\mu}{\mu + \tau}).
$$
Thus, we know that the conditional distribution for estimating $\theta$ is $\frac{N_1}{N_1 + N_2}$.
However, how can I find the conditional MLE of $\theta$? And, it seems like we don't use any information from nuisance parameters so is there any relationship between nuisance parameters and the target parameter?