Maximum Likelihood Estimator with binomial distribution question
Q. There are N fish in a lake. Six fishermen fished for 3 hours and caught 8 fish between them. They fished for another 3 hours and caught 3 more fish between them.
Assume the following model: Each fish is equally hard to catch; the probability of catching a fish is p. Let Y_1 be the amount of fish caught in the first 3 hours, Y_1 ~ B(N,p)
Let Y_2 be the amount of fish caught in the second 3 hours, Y_2 ~ B( N-Y_1, p).
Find the MLE of N and p.
Hi everyone, please could you take a look at what I've done so far:
Let Ξ = L(N)/L(N-1), ...where L(N) = (N c Y).p^Y.(1-p)^(N-Y)
= N(Y-p)/(N-1), ≥1
Then N ≤ Y/p
So for Y_1 = 8, N ≤ 8/p
Doing the same with Y_2, we find that:
N ≤ 11/p + 8
My question is how do I get an actual value for MLE of N (and hence p) for, as it stands, I only have a range of values?