In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose there exists an unknown number of items which are sequentially numbered from 1 to N. A random sample of these items is taken and their sequence numbers observed; the problem is to estimate N from these observed numbers.
Assuming tanks are assigned sequential serial numbers starting with 1, suppose that four tanks are captured and that they have the serial numbers: 19, 40, 42 and 60. Let N equal the total number of tanks predicted to have been produced, 60 equal the highest serial number observed and 4 equal the number of tanks captured.
m = 60;(*m equal the highest serial number observed*)
k = 4;(*k equal the number of tanks captured*)
μ = (m - 1) (k - 1)/(k - 2)
σ = Sqrt[((k - 1) (m - 1) (m - k + 1))/((k - 3) (k - 2)^2)]
{μ - σ, μ + σ} // N
I want to use the Monte Carlo method to simulate the value of N with the help of MMA. If possible, find the 90% confidence interval for N.
n = 100000;
Table[Count[Table[RandomChoice[Range[m], 1], n] // Flatten,
60], {m, {60, 100, 140, 180, 220, 260}}]
Through the above method, it can be seen that when the number of tanks is 60, the probability of obtaining the maximum value of the observation is the largest, but this problem is still not well solved.