I have trouble finding the likelihood function in an applied problem involving the binomial distribution.
Given are $N$ independent random variables having identical binomial distributions with the parameters $\theta$ and $n = 3$ where $n_0$ of them take on the value $0$, $n_1$ take on the value $1$, $n_2$ take on the value $2$, and $n_3$ take on the value $3$.
Now the Method of Maximum Likelihood should be used to find a formula for estimating $\theta$.
I started off from the probability distribution function of a general binomial random variable and the derivation of the maximum likelihood estimator in the general case. However, the case is now different and I got stuck already in the beginning. How should one proceed here?
$$L(\theta|n_i\vert_{i=0}^{n}, n) = \prod_{j=0}^{n} P(X=j)^{n_j} = \prod_{j=0}^{n}\left(\binom{n}{j}\theta^{j}(1-\theta)^{n-j}\right)^{n_j} = \left(\prod_{j=0}^{n}\binom{n}{j}\right)\theta^{N}(1-\theta)^{N_0-N}$$
where $N = \sum_{j=0}^{n}jn_j$ and $N_0 = n\sum_{j=0}^{n}n_j$
$$\log L(\theta|n_i\vert_{i=0}^{n}, n) = C + N\log(\theta) + (N_0-N) \log(1-\theta)$$
where $C$ is independent of $\theta$.
Take derivative wrt $\theta$ and equate to $0$,
$$N/\theta - (N_0-N)/(1-\theta) = 0 \implies \theta = \frac{N}{N_0}$$
Take double derivative wrt $\theta$ and verify that it is negative with above value of $\theta$.