There are at most 100 identical balls in a bowl, but we don’t know their exact number. However, we know that the balls are numbered with numbers 1, 2, . . . , θ and each of the numbers from the number one to number θ is used exactly once and the $θ ≥ 1$ is an integer. We pick a single ball at random. Let Y be the number in a ball we pick at random. I have to calculate the MLE estimate for θ. Suppose that we pick a ball number 57.
My attempt:
PMF: $1/θ$
Joint PMF: $1/θ^n$ and thus MLE function would be $-n*log(θ)$ and by deriving and setting the function equal to zero, the function would achieve the maximum. However, when trying to find the maximum I get that $θ = null$ which isn't possible. Does this mean that the MLE estimator is equal to 57 so that $θ=57$?