Polarization Asymmetry with Maximum Likelihood Estimator I haveto calculate a maximum likelihood estimator for the asymmetry
$$
\alpha = \frac{n_R - n_L}{n_R + n_L}
$$
where $n_R$ and $n_L$ are Poisson distributed scattering events for left/right polarized electrons with expectation value $\nu_L$ and $\nu_R$. 
A tip we got, is to "express the probability to observe $n_R$ and $n_L$ events, in terms of $\alpha$"
I just dont really know, how to approach this problem.
 A: As you have defined it, $\alpha$ is a function of random variables and is a statistic.  However, you are asking for an estimator for $\alpha$, which doesn't make sense:  estimators estimate parameters (or functions thereof), not statistics.  A statistic can be an estimator of a parameter, but as you have not properly explained what exactly you wish to estimate, your question is ill-posed.
To illustrate this with a more familiar example, suppose I have a sample $\boldsymbol X = (X_1, \ldots, X_n)$ in which observations are IID $\operatorname{Normal}(\mu,\sigma^2)$.  Now say I construct the statistic $$S = \sum_{i=1}^n \frac{1}{X_i^2+1}.$$  Your question is analogous to asking me, "What is an MLE of $S$?"  That makes no sense.  $S$ is not a parameter.  There is nothing to estimate about $S$.  If you asked me, "What is an MLE of $\mu$, then I would say $$\hat \mu = \bar X = \frac{1}{n} \sum_{i=1}^n X_i,$$ and that has nothing to do with $S$.  But that is not the only parameter or function of a parameter I can estimate; I can also ask "What is an MLE of $\sigma^2 - \mu^2$?"
So, back to your question, it would make sense if you asked "What is the MLE of $$\alpha = \frac{\nu_R - \nu_L}{\nu_R + \nu_L},$$ where $\nu_R$, $\nu_L$ are the rate parameters of two independent Poisson random variables $n_R$ and $n_L$, respectively?"  And phrased this way, $\alpha$ is a function of unknown parameters $\nu_R$, $\nu_L$, and your goal is to construct a statistic $\hat \alpha$ that estimates $\alpha$ such that the likelihood $\mathcal L(\alpha \mid \boldsymbol n_R, \boldsymbol n_L)$ is maximized for the choice $\alpha = \hat \alpha$.
