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.

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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?