$N$ people arrive at a concert, with tickets numbered $1$ to $N$.
At the entrance, they all throw their tickets to a nearby trash can.
You pull out a ticket with some number $x$ written on it.
Based on that number, you need to guess how many people are at the concert.
What is be the best guess that you could come up with?
Apparently the answer is $x$, but I'm finding it hard to construct a probabilistic argument.
What I have in mind is that the number of people can be anything in the range $[x,\infty)$.
Then, the chances decrease as you go "further away" from $x$.
But I hardly consider this a good argument.
Any idea will be appreciated.
When you draw a ticket from the bin, you're drawing a Uniform(N) random variable.
The maximum likelihood estimate of $N$ given an observation $x$ is $\hat{N}=\arg \max_{N} (\text{Probability of observing x when the parameter is $N$})= \arg \max_{N} \begin{cases} 0 & x \leq N \\ \frac{1}{N} & x \geq N\end{cases}$, which is thus $\hat{N} = x$.
Aside: If you had multiple draws of a uniform distribution, the maximum likelihood parameter estimates are the largest and smallest tickets.