This problem is simply stated but has me stumped:

In WW2 the Allies noticed that the German tanks had sequential serial numbers on them. They were able to capture some tanks, and note their serial numbers.

The problem is this: given m serial numbers from captured tanks, what is a suitable estimator for N, the total number of German tanks?

I think I can assume that N is so large and m so small that I can assume that tanks were drawn

**with replacement**(is this assumption recommended?) In this case, samples are independent.

Hence the probability of choosing any particular tank is 1/N. The probability distribution is just a uniform distribution (but a discrete distribution). The cumulative probability of choosing a tank with a number less than or equal to X is F(X) = X/N.

This is how far I've got in the solution: let the observed m tank serial numbers be x1, x2, ..., xm. Let Xmax be the random variable which represents the maximum serial number and xmax be the observed maximum number. I now try to find the probability distribution of Xmax. The probability that Xmax is less than or equal to any value x is F(x)^m = (X/N)^m (let's call this Fmax(x)).

I can find the probability that Xmax=x by simply taking the difference: P(Xmax=x) = Fmax(x)-Fmax(x-1).

Now I don't know how to proceed. Somehow I need to get an equation which features N, rearrange it so I can solve for N. I thought about taking the mean of P(Xmax), but I'm not sure that would work.

Can anybody please give me a direction for pursuing this problem?