Sampling numbers from a discrete uniform distribution where is unknown. Let be the largest number of the samples. The problem is estimating .
Solution under the Copernican principle
To explain the basic idea, let , then . Because , we will get .
An another example, , then .
Generalize the idea
The probability of all samples are not greater than ,
where is the largest number in , and is an arbitrary number greater than . Then,
Also, , then,
Examples of expectations and variances:
The wolfram code to calculate the expectation and the variance.