I'm not exactly sure how to go about this question.
Among $N$ $i.i.d.$ observations on a $Binomial(2,p)$ random variable, $N_0$ takes on the value 0, $N_1$ takes on the value 1 and $N_2$ take on the value 2. What is the maximum likelihood estimator (MLE) of $p$?
a) $\widetilde p = (N_1 + N_2)/N$
b) $\widetilde p = (2N_1 + N_2)/2N$
c) $\widetilde p = (2N_1 + N_2)/N$
d) $\widetilde p = (N_1 + 2N_2)/2N$
e) $\widetilde p = (N_1 + 2N_2)/N$
I've tried to complete the question but I'm not sure how to do it.
Thank you appreciate your help.
With $X_i \sim \mathsf{Binom}(2, p)$ and $N$ independent observations $X_1, X_2, \dots, X_n$, the MLE of $p$ is $\hat p = \frac{S}{2N},$ where $S \sim \mathsf{Binom}(2N, p)$ is the number of Successes in the $2N$ trials. If I understand your notation correctly, then the number of Successes is $S =N_1 + 2N_2.$