# Copernican principle

## Problem

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

### Basic idea

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,

where

Examples of expectations and variances:

The wolfram code to calculate the expectation and the variance.