# Copernican principle

## Problem

Sampling $m$ numbers $X = \{x_1, \dots, x_m\}$from a discrete uniform distribution $U(1,N)$ where $N$ is unknown. Let $k$ be the largest number of the $m$ samples. The problem is estimating $N$.

## Solution under the Copernican principle

### Basic idea

To explain the basic idea, let $m = 1$, then $k = x_1$. Because $% $, we will get $% $.

An another example, $% $, then $P(n > 2k) = 1/2$.

### Generalize the idea

The probability of all samples are not greater than $kn/y$,

where $k$ is the largest number in $X$, and $y$ is an arbitrary number greater than $k$. Then,

Also, $P(n = y) = P(n \ge y) - P(n \ge y + 1)$, then,

where

Examples of expectations and variances:

The wolfram code to calculate the expectation and the variance.